Thank you. Somehow I thought I could just get past my past disordered eating. That quitting drinking and embracing self acceptance would mean I no longer would have all these thoughts of fitting clothes, and what to eat and when.
And maybe I have had a few months of comfort in my body. Which is a great thing.
But recently I've been not eating. I may say I'm not hungry, but it feels like a lie. I'm hoping maybe not eating will get me back into my summer shorts. Will ensure no one comments on my weight when I go home in the summer. Will help somehow.
Reblogged this on Quintersectionality O.
The nail was once again hit on the head. I so totally relate. I'm close to facing the possibility of no more therapy, or possibly reduced therapy, and, (gasp) having to join a group. This is so terrifying to me, I can't even begin to describe it. Then, I was a good girl, and signed up for insurance after a period of time that I considered myself being civilly disobedient. Fall in line. step, step. But now I have to find a new doctor. One who will tell me that I have failed my body mass index. That, "you know you would feel better if you lost some weight' or quote me the myriad of statistics that I could become, without her knowing, that I almost became one on the other end of the ED spectrum. I don't want to have to explain it all to someone who may not (or may if given the chance) really give a crap about my 3 1/2 year struggle of not eating,losing, eating, gaining, and trying to figure out how to be kind to myself and carry on, keep moving forward, and find peace. Within the last week, just the thought of a new doctor has me not wanting to eat. I need to lose weight, before I go to a new doctor. I'm waiting for the "wow I could have had a V8 headslap" to knock me to my senses. That of course won't come.
It is a daily struggle and I appreciate your intrinsic wisdom and honesty. Thank you for your posts.


Published 11/13/2018 . Last updated 1/12/2019 .
Chocolate pumpkin pie is the pumpkin pie for chocolate lovers. Each bite of chocolate pumpkin pie brings you intense chocolate plus classic fall pumpkin pie goodness.
Chocolate pumpkin pie has a layer of rich flourless chocolate cake and a layer of pumpkin pie. Bonus: If you use a gluten-free pie crust, the whole thing is gluten-free.
To make the chocolate pumpkin pie, you spread a layer of flourless chocolate cake batter onto the pie crust.
Then, you top it with pumpkin pie batter.
Did you happen to notice that in the photo of the baked pie, the chocolate is at the top, not the bottom?
During the bake, the chocolate layer will magically float to the top, leaving the pumpkin layer at the bottom. It's a fun surprise!
Pumpkin pie always feels like it lacks a little decorative pizzazz to me. Some people decorate it whipped cream, additional pieces of pie crust cut into fall shapes, or with a caramel or chocolate drizzle.
I like to decorate the pie with chocolate and sprinkle-covered pecans.
I like using these bars for the chocolate-covered pecans.
I like to use dark chocolate in the pie itself, but use a different chocolate for the chocolate-covered nuts.
One of my favorite chocolates to use for the nuts is salted caramel chocolate. It's such a great complementing flavor.
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Chocolate pumpkin pie is the pumpkin pie for chocolate lovers. It's a traditional pumpkin pie baked right on top of a layer of flourless chocolate cake.
Combine the water, salt, sugar, and brown sugar in a small saucepan over medium heat. Stir until completely dissolved and set aside.
Either in the top half of a double boiler or in a microwave oven, melt the chocolate.
Pour the melted chocolate into the bowl of an electric mixer.
Cut the butter into pieces and beat the butter into the chocolate, one piece at a time.
Beat in the hot sugar-water.
Slowly beat in the eggs and the egg white, one at a time.
Spread the batter into the prepared pie crust.
In a medium-sized mixing bowl, mix sugar, salt, cinnamon, ginger, nutmeg, and cloves.
Pour over the chocolate layer of the pie.
Place pie on top of a cookie sheet (always a safe thing to do in case anything spills or overflows) and bake for 15 minutes.
Lower the oven temperature to 350 F and bake for another 30 minutes or until the pumpkin layer has a skin (if you touch it, your finger doesn't go right through it like a liquid) and, if you shake the pan, the pie is just a little jiggly, like jello.
Remove from the oven but keep the pie on the hot cookie sheet and let the whole thing cool for at least 30 minutes before decorating.
Place pecan halves on a cookie sheet lined with parchment paper.
Bake for five minutes or until lightly toasted.
Chop chocolate into small pieces and melt in a double-boiler or in the microwave in twenty second increments, stirring after each increment.
Dip half of each pecan half into melted chocolate, then dip in sprinkles and place back on parchment to dry. Tip: Only put a small amount of sprinkles in the bowl at a time. Once they get too much chocolate on them from multiple dips they won't stick to the nuts anymore and you will need to replenish.
Decoratively arrange pecans on top of the pie before serving.
The flourless chocolate cake recipe is one that I've been using for years from Allrecipes.
The pumpkin pie recipe is the classic recipe from Libby's.
Have you tried this recipe?Click here to leave a comment and rating!
Hi! I'm Stef Pollack. Welcome to Cupcake Project, where everyone from novice bakers to professional pastry chefs can find sweet and savory recipes to inspire and treasure.
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Oh my, I 'll have to make this for one of my friends, she loves chocolate pumpkin pie.
Ooh this sounds so good Stef. We're all chocoholics in my family and love pumpkin pie too. This should get rave reviews on Thanksgiving.
Let me know how it goes. I've been totally enjoying it!
What a brilliant idea to combine chocolate and pumpkin flavors in a pie. Sounds sooo good.
This pie needs to be added to my dessert menu this Holiday season! It looks so good!
What a great mash-up of my favourite flavours!
I have never had chocolate pumpkin pie but it looks and sounds Fabulous! Love the extra attention to the pecan topping.
Really? The chocolate floats to the top? That's magic and a lot of fun. I'm going to share this with my daughter, I think she would get a kick out of this. Love the chocolate and pumpkin pie idea. Thanks for this!
I love the combination between chocolate and pumpkin! Thank you for the delicious recipe!
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text:
Q: Steganography in Doom 2016 For the last couple of weeks i'v been trying to replicate the method in which Id Technologies used to hide hidden messages inside their own songs. For reference: https://youtu.be/yzFit0nldf4 .
I'm not sure if this was made by solely a software or a very smart use of heavy musical instruments, but that's what im trying to find out.
Using softwares like coagula might do a similar thing, but the brilliancy was how the coded sound was hidden. If you attempt to hear rendedered output it would sound obvious and aweful...
So, my question is: does anyone know how they did it?

redpajama_set_name: RedPajamaStackExchange


text:
© 2019 Southampton Concert Orchestra
committee@concertorchestra.com
Southampton Concert Orchestra was the first full amateur symphony orchestra in Southampton. It was founded in 1964 by Trevor Morgan, a local musician and teacher who trained at the Royal College of Music as a violinist, conductor and composer. Trevor emigrated to New Zealand in 1969 and the baton was handed over to Judith Bailey.
In 2001 Judith decided to return to her home county Cornwall and left the orchestra in the capable hands of Paul Ingram. In the same year Judith was awarded an Associate of the Royal Academy of Music.
The orchestra plays a wide range of music from light to serious to classical, and has given debut performances of works such as Concerto for Orchestra (Judith Bailey), Variations (Geoff Badley - an oboist with the orchestra) and many works by the composer Adrian Connell.
Many popular and talented musicians have played with the orchestra. Some of these to mention are David Campbell (clarinet) and his sister Jane (violin). It may be of interest to know that Graham Sheen was our principal bassoonist in his student days. Graham went on to become principle bassoonist for the Academy of St. Martin in the Fields and the English Chamber Orchestra, and then for the BBC Symphony Orchestra in London (where he is still in post). Many of our soloists are taken from within the orchestra.
Our members are a truly cosmopolitan group of young and old people drawn from all walks of life: clerks, teachers, academics, doctors, engineers and craftsmen to name but a few. We even have an ex-Bournemouth Symphony Orchestra member to keep us tapping in time.
Paul Ingram - Conductor
Paul Ingram studied at the Universities of Southampton and Leicester, reading Music and Management respectively.
He began conducting in 1995, working with the Southampton University Symphony, Pre-Term and Chamber Orchestras.
In 1997, Paul founded the Southampton University Sinfonietta and remained Music Director until 2004.
Paul has also worked with Essex Youth Orchestra, Young Sinfonia, Stoneleigh Youth Orchestra, COMA South, Showstoppers and the City of Southampton Orchestra. In 1999, he reached the final of the British Reserve Insurance Conducting Competition.
Paul currently conducts the Southampton Concert Orchestra and the professional orchestra, Nova Foresta Classical Players.
Cerys Rowcliffe - Leader
Cerys started playing the violin at the age of 7, obtaining grade 8 distinction by the time she was 16. As a keen musician, she has been involved with numerous musical ensembles since a young age, including Bedfordshire County Youth Orchestras, Bedfordshire County Youth Chamber Orchestra and Bedford Sinfonia. With these ensembles she has performed at the Royal Albert Hall and the Royal Festival Hall as well as playing at venues in France, Italy and Edinburgh. As part of the Whatmore Quartet, Cerys won the National Chamber Music Competition for Schools in the years 1997 and 1998. This provided her with the opportunity to play at St. John's Smith Square, London.
In 1999, Cerys began a music degree at Southampton University and during her time there she was an active member of the Southampton University Symphony Orchestra and Sinfonietta as well as playing in a string quartet for two years.
At present, Cerys works at Thornden School as Head of Year and music teacher. She is a member of the professional orchestra Nova Foresta Classical Players and also enjoys playing in String Quartets for weddings and functions. Cerys has been leading the Southampton Concert Orchestra for over 15 years and really enjoys the challenge of performing a huge variety of different orchestral music.
Southampton Concert Orchestra

redpajama_set_name: RedPajamaCommonCrawl


text:
Retro Rangers: The Emile Francis Memorial Ice Arena
by George Grimm | Dec 21, 2022
On Saturday December 10,2022, the city of Long Beach, New York renamed their municipal skating rink in honor of the late Emile Francis. The facility, now known as the Emile Francis Memorial Ice Arena, was built in 1973 and was used by the Rangers as their practice site until the 1980s when they moved their operation to Rye, New York.
Prior to 1964 when Emile Francis took over as General Manager, the Rangers practiced in a rink on the top floor of the Old Madison Square Garden called Iceland. It was small, dark, oddly shaped and for some strange reason, had aluminum boards. Built for figure skating and used for public sessions, it was approximately 140 feet long and 75 feet wide and one end was cut off at an angle, so there wasn't a full corner. Former Ranger Eddie Shack once said it was like playing in a garbage can.
"Sonia Heine used to train there, " recalled Emile Francis in a 2015 interview. "It was all glass on one end and the sun would come in there, they didn't even have blinds. So, if you were the goalkeeper at the other end of the ice you'd never see the puck coming. And if a player touched his skate on those tin boards the equipment guy had to take the guys skate, go down five floors, run over to the dressing room and sharpen the skate. It took forty minutes for the guy to come back up. So we couldn't practice properly. I said to myself I don't know how the hell they got by with this. To think that they had done this since 1926, and won three Stanley Cups. That's the rink that they practiced in. I don't know why whoever was running that club would have put up with that or why they never went out and built a rink of their own. Teams used to come in to play us, and they'd look at it and couldn't believe it. That's where you practice?
The first thing I did was to move the practice out of there to Commack, Long Island. That was the only place I could find. Where the Long Island Ducks played. It was about an hour's drive out of Manhattan. Then about a year after that I found a rink in New Hyde Park called Skateland. But that rink was the same size as the Garden, 185 by 85 feet, but I wanted a 200 by 85 foot rink. So I worked out of New Hyde Park for about five years or so. And then I had lived in Long Beach and I worked with the City of Long Beach to build us a rink right there."
Long Beach officials were joined by Emile's son and former NHL coach Bobby Francis as well as Ranger alumni Steve Vickers, Pete Stemkowski, Gilles Villemure, Andre Dore, Ron Greschner, as well as Nick Fotiu and Brian Mullen, who along with his brother Joey started out playing in the Metropolitan Junior Hockey League, founded by Francis in 1966.
Francis, who passed away in February at the age of 95 was also instrumental in creating an enclave for his players and their families in Long Beach.
"I didn't want the players living in Manhattan," said The Cat. "In the summer time there would be 100,000 people in Long Beach, but in the winter there were these nice homes out there completely furnished that you could rent.
So I got a real estate guy and I had all the homes lined up where the players could move their families for the winter and the people were only too glad to rent them. Rod Gilbert was the only guy I think was living in the city. They all lived out there. Plus the fact that I knew where everyone was.
We had an ideal set up. It was a small area and all the players lived within four to six blocks of one another. And their wives got to know one another and that built up a kind of camaraderie not only with the players but with their wives. It was a really tight knit group and that's what you wanted – one big happy family."
The Rangers, who sponsor the Junior Rangers as well as other youth leagues at the arena, have chipped in by donating equipment, renovating the locker rooms and adding safety netting. They also helped the city by donating $25,000 after Superstorm Sandy damaged the arena's ice-making equipment.
"Emile has meant so much to me and the Rangers organization for so long," said Brian Mullen. "It's so fantastic that we are finally doing something to keep his name and memory alive. He was a great man both on and off the ice."


text:
'use strict';

import * as assert from 'assert';
import { Cursor } from 'vs/editor/common/controller/cursor';
import { Position } from 'vs/editor/common/core/position';
import { ITextModelCreationOptions } from 'vs/editor/common/editorCommon';
import { Model } from 'vs/editor/common/model/model';
import { IMode } from 'vs/editor/common/modes';
import { TestConfiguration } from 'vs/editor/test/common/mocks/testConfiguration';
import { viewModelHelper as aViewModelHelper } from 'vs/editor/test/common/editorTestUtils';
import { CursorMove } from 'vs/editor/common/controller/cursorMoveCommands';
import { Range } from 'vs/editor/common/core/range';
import { Selection } from 'vs/editor/common/core/selection';
import { IEditorOptions } from 'vs/editor/common/config/editorOptions';
import { IViewModelHelper } from 'vs/editor/common/controller/cursorCommon';
import { CoreCommands } from 'vs/editor/common/controller/coreCommands';

suite('Cursor move command test', () => {

	let thisModel: Model;
	let thisConfiguration: TestConfiguration;
	let thisCursor: Cursor;

	setup(() => {
		let text = [
			'    \tMy First Line\t ',
			'\tMy Second Line',
			'    Third Line🐶',
			'',
			'1'
		].join('\n');

		thisModel = Model.createFromString(text);
		thisConfiguration = new TestConfiguration(null);
	});

	teardown(() => {
		thisCursor.dispose();
		thisModel.dispose();
		thisConfiguration.dispose();
	});

	test('move left should move to left character', () => {
		thisCursor = aCursor();
		moveTo(thisCursor, 1, 8);

		moveLeft(thisCursor);

		cursorEqual(thisCursor, 1, 7);
	});

	test('move left should move to left by n characters', () => {
		thisCursor = aCursor();
		moveTo(thisCursor, 1, 8);

		moveLeft(thisCursor, 3);

		cursorEqual(thisCursor, 1, 5);
	});

	test('move left should move to left by half line', () => {
		thisCursor = aCursor();
		moveTo(thisCursor, 1, 8);

		moveLeft(thisCursor, 1, CursorMove.RawUnit.HalfLine);

		cursorEqual(thisCursor, 1, 1);
	});

	test('move left moves to previous line', () => {
		thisCursor = aCursor();
		moveTo(thisCursor, 2, 3);

		moveLeft(thisCursor, 10);

		cursorEqual(thisCursor, 1, 21);
	});

	test('move right should move to right character', () => {
		thisCursor = aCursor();
		moveTo(thisCursor, 1, 5);

		moveRight(thisCursor);

		cursorEqual(thisCursor, 1, 6);
	});

	test('move right should move to right by n characters', () => {
		thisCursor = aCursor();
		moveTo(thisCursor, 1, 2);

		moveRight(thisCursor, 6);

		cursorEqual(thisCursor, 1, 8);
	});

	test('move right should move to right by half line', () => {
		thisCursor = aCursor();
		moveTo(thisCursor, 1, 4);

		moveRight(thisCursor, 1, CursorMove.RawUnit.HalfLine);

		cursorEqual(thisCursor, 1, 14);
	});

	test('move right moves to next line', () => {
		thisCursor = aCursor();
		moveTo(thisCursor, 1, 8);

		moveRight(thisCursor, 100);

		cursorEqual(thisCursor, 2, 1);
	});

	test('move to first character of line from middle', () => {
		thisCursor = aCursor();
		moveTo(thisCursor, 1, 8);
		moveToLineStart(thisCursor);
		cursorEqual(thisCursor, 1, 1);
	});

	test('move to first character of line from first non white space character', () => {
		thisCursor = aCursor();
		moveTo(thisCursor, 1, 6);

		moveToLineStart(thisCursor);

		cursorEqual(thisCursor, 1, 1);
	});

	test('move to first character of line from first character', () => {
		thisCursor = aCursor();
		moveTo(thisCursor, 1, 1);

		moveToLineStart(thisCursor);

		cursorEqual(thisCursor, 1, 1);
	});

	test('move to first non white space character of line from middle', () => {
		thisCursor = aCursor();
		moveTo(thisCursor, 1, 8);

		moveToLineFirstNonWhiteSpaceCharacter(thisCursor);

		cursorEqual(thisCursor, 1, 6);
	});

	test('move to first non white space character of line from first non white space character', () => {
		thisCursor = aCursor();
		moveTo(thisCursor, 1, 6);

		moveToLineFirstNonWhiteSpaceCharacter(thisCursor);

		cursorEqual(thisCursor, 1, 6);
	});

	test('move to first non white space character of line from first character', () => {
		thisCursor = aCursor();
		moveTo(thisCursor, 1, 1);

		moveToLineFirstNonWhiteSpaceCharacter(thisCursor);

		cursorEqual(thisCursor, 1, 6);
	});

	test('move to end of line from middle', () => {
		thisCursor = aCursor();
		moveTo(thisCursor, 1, 8);

		moveToLineEnd(thisCursor);

		cursorEqual(thisCursor, 1, 21);
	});

	test('move to end of line from last non white space character', () => {
		thisCursor = aCursor();
		moveTo(thisCursor, 1, 19);

		moveToLineEnd(thisCursor);

		cursorEqual(thisCursor, 1, 21);
	});

	test('move to end of line from line end', () => {
		thisCursor = aCursor();
		moveTo(thisCursor, 1, 21);

		moveToLineEnd(thisCursor);

		cursorEqual(thisCursor, 1, 21);
	});

	test('move to last non white space character from middle', () => {
		thisCursor = aCursor();
		moveTo(thisCursor, 1, 8);

		moveToLineLastNonWhiteSpaceCharacter(thisCursor);

		cursorEqual(thisCursor, 1, 19);
	});

	test('move to last non white space character from last non white space character', () => {
		thisCursor = aCursor();
		moveTo(thisCursor, 1, 19);

		moveToLineLastNonWhiteSpaceCharacter(thisCursor);

		cursorEqual(thisCursor, 1, 19);
	});

	test('move to last non white space character from line end', () => {
		thisCursor = aCursor();
		moveTo(thisCursor, 1, 21);

		moveToLineLastNonWhiteSpaceCharacter(thisCursor);

		cursorEqual(thisCursor, 1, 19);
	});

	test('move to center of line not from center', () => {
		thisCursor = aCursor();
		moveTo(thisCursor, 1, 8);

		moveToLineCenter(thisCursor);

		cursorEqual(thisCursor, 1, 11);
	});

	test('move to center of line from center', () => {
		thisCursor = aCursor();
		moveTo(thisCursor, 1, 11);

		moveToLineCenter(thisCursor);

		cursorEqual(thisCursor, 1, 11);
	});

	test('move to center of line from start', () => {
		thisCursor = aCursor();
		moveToLineStart(thisCursor);

		moveToLineCenter(thisCursor);

		cursorEqual(thisCursor, 1, 11);
	});

	test('move to center of line from end', () => {
		thisCursor = aCursor();
		moveToLineEnd(thisCursor);

		moveToLineCenter(thisCursor);

		cursorEqual(thisCursor, 1, 11);
	});

	test('move up by cursor move command', () => {
		thisCursor = aCursor();

		moveTo(thisCursor, 3, 5);
		cursorEqual(thisCursor, 3, 5);

		moveUp(thisCursor, 2);
		cursorEqual(thisCursor, 1, 5);

		moveUp(thisCursor, 1);
		cursorEqual(thisCursor, 1, 1);
	});

	test('move up by model line cursor move command', () => {
		thisCursor = aCursor();

		moveTo(thisCursor, 3, 5);
		cursorEqual(thisCursor, 3, 5);

		moveUpByModelLine(thisCursor, 2);
		cursorEqual(thisCursor, 1, 5);

		moveUpByModelLine(thisCursor, 1);
		cursorEqual(thisCursor, 1, 1);
	});

	test('move down by model line cursor move command', () => {
		thisCursor = aCursor();

		moveTo(thisCursor, 3, 5);
		cursorEqual(thisCursor, 3, 5);

		moveDownByModelLine(thisCursor, 2);
		cursorEqual(thisCursor, 5, 2);

		moveDownByModelLine(thisCursor, 1);
		cursorEqual(thisCursor, 5, 2);
	});

	test('move up with selection by cursor move command', () => {
		thisCursor = aCursor();

		moveTo(thisCursor, 3, 5);
		cursorEqual(thisCursor, 3, 5);

		moveUp(thisCursor, 1, true);
		cursorEqual(thisCursor, 2, 2, 3, 5);

		moveUp(thisCursor, 1, true);
		cursorEqual(thisCursor, 1, 5, 3, 5);
	});

	test('move up and down with tabs by cursor move command', () => {
		thisCursor = aCursor();

		moveTo(thisCursor, 1, 5);
		cursorEqual(thisCursor, 1, 5);

		moveDown(thisCursor, 4);
		cursorEqual(thisCursor, 5, 2);

		moveUp(thisCursor, 1);
		cursorEqual(thisCursor, 4, 1);

		moveUp(thisCursor, 1);
		cursorEqual(thisCursor, 3, 5);

		moveUp(thisCursor, 1);
		cursorEqual(thisCursor, 2, 2);

		moveUp(thisCursor, 1);
		cursorEqual(thisCursor, 1, 5);
	});

	test('move up and down with end of lines starting from a long one by cursor move command', () => {
		thisCursor = aCursor();

		moveToEndOfLine(thisCursor);
		cursorEqual(thisCursor, 1, 21);

		moveToEndOfLine(thisCursor);
		cursorEqual(thisCursor, 1, 21);

		moveDown(thisCursor, 2);
		cursorEqual(thisCursor, 3, 17);

		moveDown(thisCursor, 1);
		cursorEqual(thisCursor, 4, 1);

		moveDown(thisCursor, 1);
		cursorEqual(thisCursor, 5, 2);

		moveUp(thisCursor, 4);
		cursorEqual(thisCursor, 1, 21);
	});

	test('move to view top line moves to first visible line if it is first line', () => {
		let viewModelHelper = aViewModelHelper(thisModel);
		viewModelHelper.getCompletelyVisibleViewRange = () => new Range(1, 1, 10, 1);
		thisCursor = aCursor(viewModelHelper);

		moveTo(thisCursor, 2, 2);
		moveToTop(thisCursor);

		cursorEqual(thisCursor, 1, 6);
	});

	test('move to view top line moves to top visible line when first line is not visible', () => {
		let viewModelHelper = aViewModelHelper(thisModel);
		viewModelHelper.getCompletelyVisibleViewRange = () => new Range(2, 1, 10, 1);
		thisCursor = aCursor(viewModelHelper);

		moveTo(thisCursor, 4, 1);
		moveToTop(thisCursor);

		cursorEqual(thisCursor, 2, 2);
	});

	test('move to view top line moves to nth line from top', () => {
		let viewModelHelper = aViewModelHelper(thisModel);
		viewModelHelper.getCompletelyVisibleViewRange = () => new Range(1, 1, 10, 1);
		thisCursor = aCursor(viewModelHelper);

		moveTo(thisCursor, 4, 1);
		moveToTop(thisCursor, 3);

		cursorEqual(thisCursor, 3, 5);
	});

	test('move to view top line moves to last line if n is greater than last visible line number', () => {
		let viewModelHelper = aViewModelHelper(thisModel);
		viewModelHelper.getCompletelyVisibleViewRange = () => new Range(1, 1, 3, 1);
		thisCursor = aCursor(viewModelHelper);

		moveTo(thisCursor, 2, 2);
		moveToTop(thisCursor, 4);

		cursorEqual(thisCursor, 3, 5);
	});

	test('move to view center line moves to the center line', () => {
		let viewModelHelper = aViewModelHelper(thisModel);
		viewModelHelper.getCompletelyVisibleViewRange = () => new Range(3, 1, 3, 1);
		thisCursor = aCursor(viewModelHelper);

		moveTo(thisCursor, 2, 2);
		moveToCenter(thisCursor);

		cursorEqual(thisCursor, 3, 5);
	});

	test('move to view bottom line moves to last visible line if it is last line', () => {
		let viewModelHelper = aViewModelHelper(thisModel);
		viewModelHelper.getCompletelyVisibleViewRange = () => new Range(1, 1, 5, 1);
		thisCursor = aCursor(viewModelHelper);

		moveTo(thisCursor, 2, 2);
		moveToBottom(thisCursor);

		cursorEqual(thisCursor, 5, 1);
	});

	test('move to view bottom line moves to last visible line when last line is not visible', () => {
		let viewModelHelper = aViewModelHelper(thisModel);
		viewModelHelper.getCompletelyVisibleViewRange = () => new Range(2, 1, 3, 1);
		thisCursor = aCursor(viewModelHelper);

		moveTo(thisCursor, 2, 2);
		moveToBottom(thisCursor);

		cursorEqual(thisCursor, 3, 5);
	});

	test('move to view bottom line moves to nth line from bottom', () => {
		let viewModelHelper = aViewModelHelper(thisModel);
		viewModelHelper.getCompletelyVisibleViewRange = () => new Range(1, 1, 5, 1);
		thisCursor = aCursor(viewModelHelper);

		moveTo(thisCursor, 4, 1);
		moveToBottom(thisCursor, 3);

		cursorEqual(thisCursor, 3, 5);
	});

	test('move to view bottom line moves to first line if n is lesser than first visible line number', () => {
		let viewModelHelper = aViewModelHelper(thisModel);
		viewModelHelper.getCompletelyVisibleViewRange = () => new Range(2, 1, 5, 1);
		thisCursor = aCursor(viewModelHelper);

		moveTo(thisCursor, 4, 1);
		moveToBottom(thisCursor, 5);

		cursorEqual(thisCursor, 2, 2);
	});

	function aCursor(viewModelHelper?: IViewModelHelper): Cursor {
		return new Cursor(thisConfiguration, thisModel, viewModelHelper || aViewModelHelper(thisModel), false);
	}

});

interface ICursorOpts {
	text: string[];
	mode?: IMode;
	modelOpts?: ITextModelCreationOptions;
	editorOpts?: IEditorOptions;
}

// Move command

function move(cursor: Cursor, args: any) {
	CoreCommands.CursorMove.runCoreEditorCommand(cursor, args);
}

function moveToLineStart(cursor: Cursor) {
	move(cursor, { to: CursorMove.RawDirection.WrappedLineStart });
}

function moveToLineFirstNonWhiteSpaceCharacter(cursor: Cursor) {
	move(cursor, { to: CursorMove.RawDirection.WrappedLineFirstNonWhitespaceCharacter });
}

function moveToLineCenter(cursor: Cursor) {
	move(cursor, { to: CursorMove.RawDirection.WrappedLineColumnCenter });
}

function moveToLineEnd(cursor: Cursor) {
	move(cursor, { to: CursorMove.RawDirection.WrappedLineEnd });
}

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redpajama_set_name: RedPajamaGithub


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text:
No. 1 pick Mankins agrees to a contract
FOXBORO — The Patriots signed offensive lineman Logan Mankins yesterday, becoming the first NFL team to sign a player taken in the first round of this year's draft.
Mankins' signing brings all seven of the team's draft choices under contract as the Super Bowl champions prepare for their first full-squad workout of training camp on Friday.
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Rookies, quarterbacks and veterans who ended the 2004 season on injured reserve reported to training camp Sunday at Gillette Stadium.
(Published: July 26, 2005)


text:
\section{Introduction}\label{s1}
Here, for ${\lambda}\in(0,{\infty})$, we consider   the stochastic differential equation
\begin{equation}\label{e1.1}
\begin{array}{ll}
dX(t)+A(t)X(t)dt+{\lambda} X(t)dt\ni X(t)dW_t,\ t\in(0,T),\vspace*{1,5mm}\\
X(0)=x\in H,\end{array}
\end{equation}in a real Hilbert spaced $H$ whose elements are generalized functions on a bounded domain $\mathcal{O}\subset{\mathbb{R}}^d$ with a smooth boundary ${\partial}\mathcal{O}$. In examples, we have in mind that $H$ is e.g.  $L^2(\mathcal{O})$ or $H^1_0(\mathcal{O})$, $H^1(\mathcal{O})$, $H^{-1}(\mathcal{O})$.

The norm of $H$ is denoted by $|\cdot|_H$, its  scalar product by $(\cdot,\cdot)$ and its Borel $\sigma$-algebra by ${\mathcal{B}}(H)$.

$W$ is a Wiener process of the form
\begin{equation}\label{e1.2}
W(t,\xi)=\sum^{\infty}_{j=1}\mu_je_j(\xi)\beta_j(t),\ \xi\in\mathcal{O},\ t\ge0,\end{equation}
where $\{\beta_j\}^{\infty}_{j=1}$ is an independent system of real $(\mathcal{F}_t)$-Brownian motions on a probability space $\{{\Omega},\mathcal{F},\mathbb{P}\}$ with natural filtration $(\mathcal{F}_t)_{t\ge0}$ and $\{e_j\}$ is an orthonormal basis in $H$ such that both  $c_j$ and $e^2_j$, $j\in{\mathbb{N}}$, are multipliers in $H$, while $\mu_j\in{\mathbb{R}},$ $j=1,2,...$, satisfy  \eqref{e1.9} below.

As regards the nonlinear (multivalued) operator $A=A(t,{\omega}):H\to H$, the following hypotheses  will be assumed below.
\begin{itemize}\item[(i)] {\it Let ${\varphi}:[0,T]\times H\times{\Omega}\to \overline{\mathbb{R}}=]-{\infty},+{\infty}]$ be  convex lower semicontinuous in $y\in H$ and progressively measurable, i.e., for each $t\in[0,T]$ the  function ${\varphi}$ restricted to $[0,t]\times H\times{\Omega}$ is ${\mathcal{B}}([0,t])\otimes{\mathcal{B}}(H)\otimes\mathcal{F}_t$ measurable, and let
\begin{equation}\label{e1.3}
A(t,{\omega})={\partial}{\varphi}(t,{\omega}),\ \forall (t,{\omega})\in[0,T]\times{\Omega}.\end{equation}
 In particular, $y\to A(t,{\omega},y)$ is maximal monotone in $H\times H$ for all $(t,{\omega})\in[0,T]\times{\Omega}$. Furthermore, ${\varphi}$ is such that there exists $\alpha\in L^2([0,T]\times{\Omega};H)$ and $\beta\in L^2([0,T]\times{\Omega})$ such that

 {${\varphi}(t,y,{\omega})\ge(\alpha(t,{\omega}),y)-\beta(t,{\omega})\mbox{ for }dt\otimes\mathbb{P}-a.e.,\ (t,{\omega})\in[0,T]\times{\Omega}.$}}

\item[(ii)] {\it $e^{\pm W(t)}$ is a  multiplier in $H$ such that there is an $(\mathcal{F}_t)_{t\ge0}$-adapted ${\mathbb{R}}_+$-valued process $Z(t),$ $t\in[0,T]$, with
\begin{eqnarray}
&\begin{array}{rcll}
\sup\limits_{t\in[0,T]}|Z(t)| &<&{\infty},&\mathbb{P}\mbox{-a.s.},\vspace*{1,5mm}\\
|e^{\pm W(t)}y|_H&\le&Z(t)|y|_H,&\forall  t\in[0,T],\ y\in H,\end{array}\label{e1.4}\\[0,2mm]
&t\to e^{\pm W(t)}\in L(H,H)\mbox{ is continuous.}\label{e1.5} \end{eqnarray}}
\end{itemize}

 Recall that a multivalued mapping $A:D(A)\subset H\to H$ is said to be maximal monotone if it is monotone, that is, for $u_1,u_2\in D(A)$,
$$(z_1-z_2,u_1-u_2)\ge0,\ \forall  z_i\in Au_i,\ i=1,2,$$and the range $R({\lambda} I+A)$ is all of $H$ for each ${\lambda}>0$.

If ${\varphi}:H\to \overline {\mathbb{R}}$ is a convex, lower semicontinuous function, then its sub\-dif\-fe\-rential ${\partial}{\varphi}:H\to H$
\begin{equation}\label{e1.6}
{\partial}{\varphi}(u)=\{v\in H;\ {\varphi}(u)\le{\varphi}(\bar u)+(v,u-\bar u),\ \forall \bar u\in H\}\end{equation}is maximal monotone (see, e.g., \cite{1}).

The conjugate ${\varphi}^*$ of $H$ defined by
\begin{equation}\label{e1.7}
{\varphi}^*(v)=\sup\{(u,v)-{\varphi}(u);\ u\in H\}
\end{equation}satisfies
\begin{equation}\label{e1.8}
\begin{array}{rcll}
{\varphi}(u)+{\varphi}^*(v)&\ge&(u,v),&\forall  u,v\in H,\vspace*{1,5mm}\\
{\varphi}(u)+{\varphi}^*(v)&=&(u,v),&\mbox{iff }v\in{\partial}{\varphi}(u).\end{array}
\end{equation}

As regards the basis $\{e_j\}$ arising in the definition of the Wiener process~$W$, we assume also that, for the multipliers $e^2_j$, we have
\begin{itemize}\item[(iii)] {\it For ${\gamma}_j=\max\{\sup\{|ue_j|_H;\ |u|_H=1\},(\sup\{ u e^2_j|_H;\,|u|_H=1\})^{\frac12},1\}$, we assume
\begin{equation}\label{e1.9}
\nu=\sum^{\infty}_{j=1}\mu^2_j{\gamma}^2_j<{\infty},
\end{equation}and that ${\lambda}>\nu$. }\end{itemize}

Clearly, then
\begin{equation}\label{e1.10}
\mu=\frac12\sum^{\infty}_{j=1}\mu^2_je^2_j
\end{equation}is a  multiplier in $H$.

It should be noted that the condition ${\lambda}>\nu$ in \eqref{e1.9} is made only for   convenience. In fact, by the substitution $X\to\exp(-{\lambda} t)X$ and replacing $A(t)$ by $u\to e^{-{\lambda} t}A(t)(e^{{\lambda} t}u)$ we can always change ${\lambda}$ in \eqref{e1.1} to a big enough ${\lambda}$ which satisfies ${\lambda}>\nu$.   It should be emphasized that a general existence and uniqueness result for equation \eqref{e1.1} is known only for the special case where $A(t)$ are monotone and demicontinuous operators from $V$ to $V'$, where $(V,V')$ is a pair of reflexive Banach spaces in duality with the Hilbert space $H$ as pivot space, that is,  $V\subset H (\equiv H')\subset V'$ densely and continuously.  If, in addition, for ${\alpha}_1\in(0,{\infty})$, ${\alpha}_2,{\alpha}_3\in{\mathbb{R}}$,
\begin{eqnarray}
{}_{V'}(A(t)u,u)_{V}&\ge&\alpha_1\|u\|^p_V+\alpha_2|u|^2_H,\ \forall  u\in V,\label{e1.11}\\[1mm]
\|A(t)u\|_{V'}&\le&\alpha_3\|u\|^{p-1}_V,\ \forall  u\in V, \label{e1.12}
\end{eqnarray}where $1<p<{\infty}$, then equation \eqref{e1.1} has under assumptions (i)--(iii)  a unique strong solution $X\in L^p((0,T)\times{\Omega};V)$ (see \cite{8}, \cite{13-b}, \cite{9}, \cite{10}). We~noted before that  assumption (i) implies that $A(t,{\omega})$ is maximal monotone in $H$ for all $t\in[0,T]$, though not every maximal monotone operator $A(t):D(A(t))\subset H\to H$ has a realization in a convenient pair of spaces $(V,V')$ such that \eqref{e1.8}--\eqref{e1.9} hold.
Though assumptions \eqref{e1.11}--\eqref{e1.12} hold for a large class of stochastic parabolic equations in Sobolev spaces $W^{1,p}(\mathcal{O}),$ $1\le p<{\infty}$ (see \cite{4}), some other important stochastic PDEs are not covered by this functional scheme. For instance, the variational stochastic differential equations, nonlinear parabolic stochastic equations in $W^{1,1}(\mathcal{O})$, in Orlicz-Sobolev spaces on $\mathcal{O}$  or in $BV(\mathcal{O})$ (bounded variation stochastic flows) cannot be treated in this functional setting. As a matter of fact, contrary to what happens for deterministic infinite differential equations, there is no general existence theory for equation \eqref{e1.1} under assumption (i)--(iii). The definition of a convenient concept of a weak solution to be unique and continuous with respect to data is a challenging objective of the existence   theory of the infinite dimensional SDE. In this paper, we introduce such a  solution $X$ for \eqref{e1.1} which is defined as a minimum point of a certain convex functional defined on a suitable space of $H$-valued processes on $(0,T)$. This idea was developed in \cite{6}  for nonlinear operators $A(t):V\to V'$ satisfying condition \eqref{e1.11}--\eqref{e1.12} and is based on the so-called {\it Brezis--Ekeland variational principle} \cite{6}. Such a solution  in the sequel will be called the {\it variational solution} to \eqref{e1.1}. (Along these lines see also \cite{2}, \cite{3}, \cite{4b}, \cite{4}.)

\section{The variational solution to equation \eqref{e1.1}}
\setcounter{equation}{0}

First, we transform  equation \eqref{e1.1} into a random differential equation via the substitution
\begin{equation}\label{e2.1}
X(t)=e^{W(t)}(y(t)+x),\ t\in[0,T],
\end{equation}
which, by It\^o's product rule,

$$dX=e^Wdy+e^W(y+x)dW+\mu e^W(y+x)dt,$$leads to
\begin{equation}\label{e2.2}
\begin{array}{l}
\displaystyle\frac{dy(t)}{dt}+e^{-W(t)}
A(t)(e^{W(t)}(y(t)+x))+(\mu+{\lambda})(y(t)+x)\ni 0,\\\hfill t\in(0,T),\vspace*{1,5mm}\\
y(0)=0.\end{array}
\end{equation}
(In the following, we shall omit ${\omega}$ from the notation $A(t,{\omega})$.)

 As a matter of fact, the equivalence between \eqref{e1.1} and \eqref{e2.2} is true only for  a smooth solution $y$ to \eqref{e2.2}, that is, for pathwise absolutely continuous strong solutions to \eqref{e2.2} (see \cite{5}, \cite{5a}). In the sequel, we shall define a generalized (variational) solution for the random Cauchy problem \eqref{e2.2} and will call the corresponding process $X$ defined by \eqref{e2.1} {\it the variational solution to}~\eqref{e1.1}.

We shall treat equation \eqref{e2.2} by the operator method developed in \cite{5a}. Namely,   consider the space $\mathcal{H}$ of all $H$-valued processes $y:[0,T]\to H$ such that
$$|y|_{\mathcal{H}}=\({\mathbb{E}}\int^s_0|e^{W(t)}y(t)
|^2_{H}dt\)^{\frac12}<{\infty},$$
which have an $(\mathcal{F}_t)_{t\ge0}$-adapted version. Here ${\mathbb{E}}$ denotes the expectation with respect to $\mathbb{P}.$   The space $\mathcal{H}$ is a Hilbert space with the scalar product
$$\<y,z\>={\mathbb{E}}\int^s_0(e^{W(t)}y(t),e^{W(t)}z(t))dt,\ y,z\in\mathcal{H}.$$

We set $\delta=\frac12\,({\lambda}-\nu)$.
Now, consider the operators ${\mathcal{A}}:D({\mathcal{A}})\subset\mathcal{H}\to\mathcal{H}$ and ${\mathcal{B}}:D({\mathcal{B}})\subset\mathcal{H}\to\mathcal{H}$ defined by
\begin{equation}\label{e2.3}
\begin{array}{rcl}
({\mathcal{A}} y)(t)&\!\!\!=\!\!\!&e^{-W(t)}A(t)(e^{W(t)}(y(t)+x))+\delta (y+x),\ \forall  y\in D({\mathcal{A}} ),\\
&&\hfill t\in[0,T],\vspace*{1,5mm}\\
D({\mathcal{A}} )&\!\!\!=\!\!\!&\{y\in\mathcal{H} ;\ e^{W(t)}(y(t)+x)\in D(A(t)),\ \forall  t\in[0,T]\mbox{ and }\\
&&\hfill e^{-W}A(e^{W}(y+x))\in\mathcal{H} \},\end{array}
\end{equation}
\begin{equation}\label{e2.4}
\begin{array}{rcl}
({\mathcal{B}} y)(t)&\!\!\!=\!\!\!&\displaystyle\frac{dy}{dt}\,(t)+(\mu+\nu+\delta)(y+x),\ \mbox{a.e.}\ t\in(0,T),\ y\in D({\mathcal{B}} ),\vspace*{1,5mm}\\
D({\mathcal{B}} )&\!\!\!=\!\!\!&\left\{y\in\mathcal{H} ;\ y\in W^{1,2}_0([0,T];H),\ \mathbb{P}\mbox{-a.s.},\ \displaystyle\frac{dy}{dt}\in\mathcal{H} \right\}\!.  \end{array}
\end{equation}
Here, $W^{1,2}_0([0,T];H)$ denotes the space $\{y\in W^{1,2}([0,T];H);\,y(0)=0\}$, where $W^{1,2}([0,T];H)$ is the Sobolev space $\left\{y\in L^2(0,T;H),\,\frac{dy}{dt}\in L^2(0,T;H)\right\}$. We recall that $W^{1,2}([0,T];H)\subset AC([0,T];H)$,   the space of all $H$-valued absolutely continuous functions on $[0,T]$.

 Then we may rewrite equation \eqref{e2.2} as
\begin{equation}\label{e2.5}
{\mathcal{B}} y+{\mathcal{A}} y\ni0 .\end{equation}(If $A(t)$ is multivalued,   we replace $A(t)(e^W(y+x))$ in \eqref{e2.3} by $\{\eta(t);\ \eta(t)\in A(t)(e^{W(t)}(y(t)+x)),\mbox{ a.e. }(t,{\omega})\in(0,T)\times{\Omega}\}$.)

Consider the functions $\Phi :\mathcal{H} \to\overline{\mathbb{R}}$ defined by
\begin{equation}\label{e2.6}
\displaystyle\Phi (y)={\mathbb{E}}\int^T_0({\varphi}(t,e^{W(t)}(y(t)+x))+\frac\delta2\,| e^{W(t)}(y(t)+x)|^2_H)dt,\ \forall  y\in\mathcal{H} .\end{equation}It is easily seen that $\Phi $ is convex, lower-semicontinuous and
\begin{equation}\label{e2.7}
{\partial}\Phi ={\mathcal{A}}  .\end{equation}As regards the operator ${\mathcal{B}} $,  we have

\begin{lemma}\label{l2.2} For each   $y \in D({\mathcal{B}} )$ we have
\begin{equation}\label{e2.8}
\begin{array}{lcl}
\<{\mathcal{B}} y,y\> &=&\displaystyle\frac12\,{\mathbb{E}}|e^{W(T)}y(T)|^2_H+(\nu+{\delta})
|y|^2_{\mathcal{H} }\vspace*{1,5mm}\\
&&-\displaystyle\frac12\,{\mathbb{E}}\int^T_0\sum^{\infty}_{j=1}|e^Wy e_j|^2_H\mu^2_jdt\vspace*{1,5mm}\\
&\ge&\displaystyle\frac12\,{\mathbb{E}}|e^{W(T)}y(T)|^2_H
+\displaystyle\frac\lbb2\,|y|^2_{\mathcal{H} } .\end{array}\end{equation}\end{lemma}

\noindent {\bf Proof.} We have
\begin{equation}\label{e2.9}
\begin{array}{lcl}
\<{\mathcal{B}} y, y \> &=&\displaystyle{\mathbb{E}}\int^T_0\(e^{W(t)}\frac{dy}{dt}\,(t),e^{W(t)
}y(t)\)dt\vspace*{1,5mm}\\
&&\displaystyle+\,{\mathbb{E}}\int^T_0((\mu+\nu+\delta)e^W y,e^W y)dt.\end{array}\end{equation}Taking into account that
$$d(e^Wy)=e^W \,dy+e^W y\,dW+\mu e^Wy\,dt,\ \forall  y\in D({\mathcal{B}} ),$$we get via It\^o's formula that (see \cite{4})
$$\begin{array}{lcl}
\displaystyle\frac12\,d|e^Wy|^2_H&=&\(e^W\,\displaystyle\frac{dy}{dt},e^Wy\)dt+(e^Wy,e^Wy\,dW)+(\mu e^Wy,e^Wy)dt\vspace*{1,5mm}\\
&&+\displaystyle\frac12\sum^{\infty}_{j=1}\mu^2_j|e^Wye_j|^2_Hdt.\end{array}$$Hence
$$\begin{array}{lcl}
{\mathbb{E}}\displaystyle\int^T_0\(e^W\frac{dy}{dt},e^W y\)dt&=&\displaystyle\frac12\,{\mathbb{E}}|e^{W(T)}y(T)|^2_H
-{\mathbb{E}}\displaystyle\int^T_0(\mu e^W y,e^W y)dt\vspace*{1,5mm}\\
&&-\displaystyle\frac12\,{\mathbb{E}}\int^T_0\sum^{\infty}_{j=1}|e^W y e_j|^2_H\mu^2_j\,dt,\end{array}$$
and so, because ${\lambda}>\nu$, by \eqref{e1.9}, \eqref{e2.9}, we get \eqref{e2.8}, as claimed.{\hfill\rule{1,5mm}{1,5mm}}\bigskip

Consider now the conjugate $\Phi^* :\mathcal{H} \to\overline{\mathbb{R}}$ of functions
$\Phi $, that is,
$$\Phi^* (z)=\sup\{\<z,y\>_{\mathcal{H} }-\Phi (y);\ y\in\mathcal{H} \}.$$By \eqref{e2.6}, we see that (see \cite{16-b})
\begin{equation}\label{e2.10}
\Phi^* (z)={\mathbb{E}}\int^T_0(\psi^* (t,e^{W(t)}z(t))-(e^{W(t)}z(t),e^{W(t)}x))dt,\end{equation}where $\psi^*$ is the conjugate of the function
\begin{equation}\label{e2.11}
\psi(t,y)={\varphi}(t,y)+\frac\delta2\,|y|^2_H,\end{equation}
that is,
\begin{equation}\label{e2.11a}
\psi^*(t,v)=\sup\{(v,y)-{\varphi}(t,y)-
\frac\delta2\,|y|^2_H;\ y\in H\}.\end{equation}
We recall  (see \eqref{e1.8}) that
\begin{equation}\label{e2.12}
\Phi (y)+\Phi^* (u)\ge\<y,u\> ,\ \forall  y,u\in\mathcal{H} ,  \end{equation}with equality if and only if $u\in{\partial}\Phi (y)$. We infer that $y^*$ is  {\it a solution to equation \eqref{e2.5} if and only if}
\begin{equation}\begin{array}{lcl}
y^*&=&{\rm arg}\min\limits_{\hspace*{-6mm}(y,u)\in{\mathcal{D}}({\mathcal{B}})\times\mathcal{H}}\{\Phi (y)+\Phi^* (u)-\<y,u\> ;\ {\mathcal{B}}  y+u=0\}\vspace*{1,5mm}\\
&=&{\rm arg}\min\limits_{\hspace*{-6mm}(y,u)\in{\mathcal{D}}({\mathcal{B}})\times\mathcal{H}}\{\Phi (y)+\Phi^* (u)+\<{\mathcal{B}} y,y\> ;\ {\mathcal{B}} y+u=0\}\end{array}\label{e2.13}
\end{equation}and
\begin{equation}\label{e2.14}
\Phi (y^*)+\Phi^* (u^*)+\<{\mathcal{B}} y^*,y^*\> =0.\end{equation}
Taking into account \eqref{e2.10} and  recalling \eqref{e2.6}, \eqref{e2.8}, we have
 \begin{equation}\label{e2.15}
\hspace*{-4mm}\begin{array}{ll}
y^*={\rm arg}\min\limits_{\hspace*{-6mm}(y,u)\in{\mathcal{D}}({\mathcal{B}})\times\mathcal{H}}\Bigg\{\!\!\!&{\mathbb{E}}\displaystyle\int^T_0 \Big({\varphi}(t,e^{W(t)}(y(t)+x))
+\displaystyle\frac\delta2\,|e^{W(t)}(y(t)+x)|^2_H\vspace*{1,5mm}\\  &+\,\psi^*(t,e^{W(t)}u(t))-
(e^{W(t)}u(t),e^{W(t)}x)\\
&+\eta(e^{W(t)}y(t))\Big)dt
+\displaystyle\frac12\,{\mathbb{E}}|e^{W(T)}y(T)|^2_H;\ {\mathcal{B}} y+u=0\Bigg\},\end{array}\hspace*{-15mm}
\end{equation}
where
\begin{equation}\label{e2.16}
\eta(z)=(\nu+\delta)|z|^2_H-\frac12\sum^{\infty}_{j=1}|z e_j|^2_H\mu^2_j.\end{equation}
We note also that, by It\^o's product rule, we have, for $u\in\mathcal{H},$ $y\in{\mathcal{D}}({\mathcal{B}})$,
$$
\begin{array}{l}
-{\mathbb{E}}\displaystyle\int^T_0(e^{W(t)}u(t),e^{W(t)}x)dt\\
\qquad ={\mathbb{E}}\displaystyle\int^T_0\bigg(e^Wx,e^W\(\frac{dy}{dt}
+(\mu+\nu+{\delta})(y+x)\)\bigg)dt\\
\qquad
={\mathbb{E}}\displaystyle\int^T_0(e^Wx,d(e^Wy))
-\int^T_0(e^Wx,\mu e^Wy
-(\mu+\nu+{\delta})(y+x)e^W)dt\\
\qquad={\mathbb{E}}\displaystyle\int^T_0(e^Wx,(\mu+\delta+\nu)(y+x)e^W)dt\\
\qquad
+{\mathbb{E}}\displaystyle\int^T_0d(e^Wx,e^Wy)-\int^T_0
\((e^Wy,e^W(1+\mu) x)-(\mu e^Wy,e^Wx)\)dt\\
\qquad
={\mathbb{E}}(e^{W(T)}x,e^{W(T)}y(T))
-\displaystyle\int^T_0\((e^Wy, e^W(1+\mu) x)
-(\mu e^Wy,e^Wx)\)dt\\
\qquad
+{\mathbb{E}}\displaystyle\int^T_0(e^Wx,(\mu+\nu+{\delta})
(y+x)e^W)dt\end{array}$$
$$\begin{array}{l}
={\mathbb{E}}(e^{W(T)}x,e^{W(T)}y(T))
+{\mathbb{E}}\displaystyle\int^T_0(e^Wx,
((\mu+\nu+{\delta})(y+x)-\mu y)
e^W)dt\\
\qquad-\displaystyle\int^T_0(e^Wy,e^W(1+\mu)x)dt\\
\qquad
={\mathbb{E}}\displaystyle\int^T_0(e^{W}
((\nu+\delta)(y +x)+\mu x),
e^{W }x)dt
+{\mathbb{E}}(e^{W(T)}y(T),e^{W(T)}x)\\
\qquad-\displaystyle\int^T_0(e^Wy,e^W(1+\mu)x)dt.\end{array}$$

Let $\mathcal{H}_0$ denote the set of all $u\in L^2([0,T]\times{\Omega};H)$ which have an $(\mathcal{F}_t)_{t\ge0}$-adapted version.
We set, for $y\in\mathcal{H},\ u\in\mathcal{H}_0,$
\begin{eqnarray}
\!\!\!G_1(y)\!\!\!&{=}
& {\mathbb{E}}\int^T_0 {\varphi}(t,e^{W(t)}(y(t)+x))dt\label{e2.17}\\ &&+\,{\mathbb{E}}\displaystyle\int^T_0((e^{W(t)}
((\nu+\delta)(y(t)+x)
+\mu x),e^{W(t)}x)\nonumber\\
&&+\,\displaystyle\frac\delta2\,|e^{W(t)}(y(t)+x)|^2_H
+\eta(e^{W(t)}y(t)))dt\nonumber\\[1mm]
&&+\displaystyle\frac12\,{\mathbb{E}}|e^{W(T)}y(T)|^2_H
+{\mathbb{E}}(e^{W(T)}y(T),e^{W(T)}x)\nonumber\\
&&-{\mathbb{E}}\displaystyle\int^T_0(e^{W(t)}y(t),e^{W(t)}
(1+\mu)x)dt,\nonumber\\[1mm]
\!\!\!G_2(u)\!\!\!&{=}&\!\!\!{\mathbb{E}}\int^T_0 \psi^*(t,u(t))  dt,\quad\label{e2.18}
\end{eqnarray}where $\psi^*$ is given by \eqref{e2.11a}.

By \eqref{e2.15} it follows that $y^*$ is a solution to equation \eqref{e2.5} {\it if~and only if}
\begin{equation}\label{e2.19}
y^*={\rm arg}\min\limits_{(y,u)\in{\mathcal{D}}({\mathcal{B}})\times\mathcal{H}_0}\{G_1(y)+G_2(u);\ e^W {\mathcal{B}} y+u=0\}\end{equation}{\it and}
\begin{equation}\label{e2.20}
G_1(y^*)+G_2(u^*)=0.\end{equation}
It should be said, however, that under our assumptions the convex minimization problem \eqref{e2.19} might have no solution $(y^*,u^*)$ because, in general, $G_2$ is not coercive on the space $\mathcal{H}$. ($G_2$ is, however, coercive if ${\varphi}$ is bounded on bounded sets of $H$. But such a condition is too restrictive for applications to PDEs.) So, we are led to replace \eqref{e2.19} by a relaxed optimization problem to be defined below.

Let
\begin{equation}\label{e2.20a}
\mathcal{X} = L^2({\Omega};(W^{1,2}([0,T];H))'), \end{equation}
where $(W^{1,2}([0,T];H))'$ is the dual space of $W^{1,2}([0,T];H)$.

Define the operator $\widetilde {\mathcal{B}} : \mathcal{H} \times L^2({\Omega};H) \to \mathcal{X}$ by

\begin{equation}\label{e2.21}
\begin{array}{ll}
(\widetilde{\mathcal{B}}(y,y_1))(\theta) \!\!\!&=
 {\mathbb{E}}(e^{W(T)}y_1,\theta(T))
 +{\mathbb{E}}\displaystyle\int^T_0((\nu+\delta)
 (y(t)+x)\vspace*{1,5mm}\\
 &
 +\mu x)e^{W(t)},\theta(t))dt
 -{\mathbb{E}}\displaystyle\int^T_0\(e^{W(t)}y(t),
 \frac{d\theta}{dt}\,(t)\)dt,\ \vspace*{1,5mm}\\
 &\hfill\forall \theta\in L^2({\Omega};W^{1,2}([0,T];H)).\end{array}
 \end{equation}
We note that $y_1({\omega}) \in H$ can be viewed as the trace   of $y({\omega})$ at $t=T$.

Indeed, if $y\in {\mathcal{D}}({\mathcal{B}})$, we have via It\^o's formula
$$\begin{array}{lcl}
{\mathbb{E}}\displaystyle\int^T_0(e^W{\mathcal{B}} y,\theta)dt\!\!\!
&=\!\!\!&{\mathbb{E}}\(\displaystyle\int^T_0(d(e^Wy),\theta)
-\int^T_0(e^W\mu y, \theta)dt\)\vspace*{1,5mm}\\
&&
+{\mathbb{E}}\displaystyle\int^T_0(e^W(\mu+\nu+\delta)
(y+x),\theta)dt\vspace*{1,5mm}\\
&=\!\!\!&{\mathbb{E}}(e^{W(T)}y(T),\theta(T))\vspace*{1,5mm}\\
&&
+{\mathbb{E}}\displaystyle\int^T_0(e^W((y+x)(\nu+\delta)+\mu x),\theta)dt\vspace*{1,5mm}\\
&&-{\mathbb{E}}\displaystyle\int^T_0\(e^Wy,
\frac{d\theta}{dt}\)dt,\
\forall \theta\in L^2({\Omega};W^{1,2}([0,T];H)).\end{array}$$
This means that $\widetilde{\mathcal{B}}(y,y(T))=e^W{\mathcal{B}} y,$ $\forall  y\in {\mathcal{D}}({\mathcal{B}})$. We set
\begin{equation}\label{e2.23-z}
\begin{array}{ll}
\widetilde G_1(y,y_1)=\!\!\!
&{\mathbb{E}}\displaystyle\int^T_0{\varphi}
(t,e^{W(t)}(y(t)+x))dt\vspace*{1,5mm}\\
&
+{\mathbb{E}}\displaystyle\int^T_0\bigg((e^{W(t)}
((\nu+\delta)(y(t)+x)+\mu x),e^{W(t)}x) \vspace*{1,5mm}\\
&
+\displaystyle\frac\delta2\,
|e^{W(t)}(y(t)+x)|^2_H
+\eta(e^{W(t)}y(t))\bigg)dt\vspace*{1,5mm}\\
&-{\mathbb{E}}\displaystyle\int^T_0
(e^{W(t)}y(t),e^{W(t)}
(1+\mu)x)dt
+\displaystyle\frac12\,{\mathbb{E}}|e^{W(T)}y_1|^2_H\vspace*{1,5mm}\\
&+{\mathbb{E}}(e^{W(T)}y_1,e^{W(T)}x),\
 \  \forall  (y,y_1)\in\mathcal{H}\times L^2({\Omega};H)\end{array}\end{equation}
and note that $\widetilde G_1(y;y(T))=G_1(y),\ \forall  y\in{\mathcal{D}}({\mathcal{B}})$.

We note also that, if  $y_n\in{\mathcal{D}}({\mathcal{B}})$ such that $y_n\to y$ weakly in $\mathcal{H}$ and $y_n(T)\to y_1$ weakly in $L^2({\Omega};H)$, then
\begin{equation}\label{e2.23-zz}
e^W{\mathcal{B}} y_n\to\widetilde{\mathcal{B}}(y,y_1)\mbox{ weakly in }\mathcal{X}.\end{equation}

Let $\overline G:\mathcal{H}\times L^2({\Omega};H)\times \mathcal{X}\to\overline{\mathbb{R}}$ be the lower semicontinuous closure of the function $G(y,y_1,u)=\widetilde G_1(y,y_1)+G_2(u)$ in $\mathcal{H}\times L^2({\Omega};H)\times\mathcal{X}$, on the set $\{(y,y_1,u)\in\mathcal{H}\times L^2({\Omega};H)\times\mathcal{X};\ e^W{\mathcal{B}} y+u=0\},$ that is,
\begin{equation}\label{e2.22}
\begin{array}{r}
\overline G(y,y_1,u)=\liminf
\{G(z,z(T),u);\,z(T)\to y_1\mbox{ in }L^2({\Omega};H),\vspace*{1,5mm}\\
z\in{\mathcal{D}}({\mathcal{B}}),\,(z,v)\to(y,u)\mbox{ in }\mathcal{H}\times\mathcal{X};\,
e^W {\mathcal{B}} z+v=0\}. \end{array} \end{equation}
(Here and everywhere in the following, by $\to$ we mean weak
convergence.)

Taking into account that the function $\widetilde G_1$ is convex and lower semicontinuous in $\mathcal{H}\times L^2({\Omega};H)$, we have by \eqref{e2.22}
\begin{equation}\label{e2.24-b}
\begin{array}{r}  \overline G(y,y_1,u){=}\widetilde G_1(y,y_1){+}\liminf
\{G_2(v); (z,v){\to}(y,u)\mbox{ in }\mathcal{H}{\times}\mathcal{X},\vspace*{1,5mm}\\   e^W{\mathcal{B}} z{+}v{=}0\}.\end{array}
\end{equation}

Now, we relax \eqref{e2.19} to the convex minimization problem
$${\rm Min}\{\overline G(y,y_1,u);\, \widetilde{\mathcal{B}}(y,y_1)+u=0;\ (y,y_1,u)\in\mathcal{H}\times L^2({\Omega};H)\times\mathcal{X}\}.\leqno{\rm(P)}$$
We have
\begin{theorem}\label{t2.1} Let $x\in H$. Then problem {\rm(P)} has a unique solution $(y^*,y^*_1,u^*)$ $\in\!\mathcal{H}{\times } L^2({\Omega};H){\times}\mathcal{X},$ with $u^*{=}-\widetilde{\mathcal{B}}(y^*,y^*_1)$.
Moreover, ${\varphi}(\cdot,e^{W}(y^*+x))\!\in\! L^1((0,T){\times}{\Omega}).$\end{theorem}
\noindent {\bf Proof.} Let $m$ be the infimum in (P)  and let $(y_n,u_n)\in {\mathcal{D}}({\mathcal{B}})\times\mathcal{H}$ be such that
\begin{eqnarray}
&m\le G(y_n,y_n(T),u_n)\le m+\displaystyle\frac1n,\ \ \forall  n\in\mathbb{N},\label{e2.23}\\
&e^W{\mathcal{B}} y_n+u_n=0.\label{e2.24}
\end{eqnarray}
Since, by assumption (iii), for some $C_1,C_2\in]0,{\infty}[$,
  $$\widetilde G_1(y_n,y_n(T))\ge C_1(|y_n|^2_\mathcal{H}+{\mathbb{E}}|e^{W(T)}y_n(T)|^2_H)-C_2,$$
    we have along a subsequence
$$y_n\longrightarrow y^*\mbox{ weakly in }\mathcal{H},\ \ y_n(T)\to y^*_1\mbox{ weakly in }L^2({\Omega};H),$$and so, by \eqref{e2.23-zz}, we have
$$u_n\longrightarrow u^*=-\widetilde{\mathcal{B}} (y^*,y^*_1)\mbox{ weakly in }\mathcal{X}.$$

As $\overline G$ is  weakly  lower semicontinuous on $\mathcal{H}\times L^2({\Omega};H)\times\mathcal{X}$,   we see by \eqref{e2.23} that
$$\overline G(y^*,y^*_1,u^*)=m,$$as claimed.
The  uniqueness of $(y^*,y^*_1,u^*)$ is immediate because the function $\overline G(\cdot,\cdot,u)$ is  strictly convex on~$\mathcal{H}\times L^2({\Omega};H)$ for all $u\in\mathcal{X}$.{\hfill\rule{1,5mm}{1,5mm}}

\begin{definition}\label{d2.2}\rm A pair $(y^*,y^*_1)$ such that  $(y^*,y^*_1,u^*)\in\mathcal{H}\times L^2({\Omega};H)\times\mathcal{X}$,  $u^*=-\widetilde B(y^*,y^*_1)$, is a solution to problem (P), is called {\it the variational solution to equation} \eqref{e2.2}, and $X^*=e^W(y^*+x)$ is called the {\it variational solution to equation} \eqref{e1.1}.\end{definition}

The variational solution $X^*:(0,T)\to H$ is an $(\mathcal{F}_t)_{t\ge0}$-adapted process.

 Theorem \ref{t2.1} can be rephrased as:
 \begin{theorem}\label{t2.4-b} Under hypotheses {\rm(i)--(iii)}, equation  \eqref{e1.1} has a unique va\-ria\-tio\-nal solution $X^*\in L^2((0,T)\times{\Omega};H)$ with ${\varphi}(t,X^*)\in L^1((0,T)\times{\Omega}).$
 \end{theorem}
 It should be noted that $y^*$ and $X^*$, as well, are not pathwise continuous on $[0,T]$. As seen later on, this happens, however, in some specific cases with respect to a weaker topology.

In the next section, we shall see how problem (P) looks like in a few important examples of stochastic PDEs.
\begin{remark}\rm The above formulation of the variational solution $X^*$ is strongly dependent on the subdifferential form \eqref{e1.3} of the operator $A(t)$.  The extension of the above technique to a general maximal monotone function $A(t):H\to H$ remains to be done using the Fitzpatrick formalism (see~\cite{11}).\end{remark}
\section{Nonlinear parabolic stochastic differential equations}
\setcounter{equation}{0}
We consider here the stochastic differential equation
\begin{equation}\label{e3.1}
\begin{array}{l}
dX-{\rm div}_\xi(a(t,\nabla X))dt+{\lambda} X\,dt=X\,dW\mbox{ in }(0,T)\times\mathcal{O},\vspace*{1,5mm}\\
X=0\mbox{ on }(0,T)\times{\partial}\mathcal{O},\vspace*{1,5mm}\\
X(0,\xi)=x(\xi),\ \xi\in\mathcal{O}\subset{\mathbb{R}}^d,\end{array}\end{equation}
where $x\in H$,  $W$ is the Wiener process \eqref{e1.2} in $H=L^2(\mathcal{O})$, $\mathcal{O}$ is a bounded and open subset of ${\mathbb{R}}^d$ with smooth boundary ${\partial}\mathcal{O}$, and  $a:(0,T)\times{\mathbb{R}}^d\to{\mathbb{R}}^d$ is a nonlinear mapping of the form
\begin{equation}\label{e3.2}
a(t,z)={\partial}_z j(t,z),\ \forall  z\in{\mathbb{R}}^d,\ t\in[0,T],\end{equation}
where $j:(0,T)\times{\mathbb{R}}^d\to{\mathbb{R}}$ is measurable, convex, lower semicontinuous in $z$ and
\begin{eqnarray}
\lim_{|z|\to{\infty}}\frac{j(t,z)}{|z|}&=&+{\infty},\ \ t\in[0,T],\label{e3.3}\\[1mm]
\lim_{|v|\to{\infty}}\frac{j^*(t,v)}{|v|}&=&+{\infty},\ \ t\in[0,T],\label{e3.4}
\end{eqnarray}uniformly with respect to $t\in[0,T].$

We note that, if the function  $(t,y)\to j(t,y)$ is bounded on bounded subsets of $[0,T]\times{\mathbb{R}}^d$, then \eqref{e3.4} automatically holds by the conjugacy formula \eqref{e1.8}, that is,
$$j^*(t,v)\ge v\cdot z-j(t,z),\ \forall  v,z\in {\mathbb{R}}^d,\ t\in[0,T].$$

It should be noted that equation \eqref{e3.1} cannot be treated in the functional setting \eqref{e1.11}-\eqref{e1.12} which require polynomial growth and boundedness  for $j(t,\cdot)$, while assumptions \eqref{e3.3}--\eqref{e3.4} allow nonlinear diffusions $a$ with slow growth  to $+{\infty}$ as well as superlinear growth  of the form
$$a(t,z)=a_0\exp(a_1|z|^p{\rm sgn}\ z).$$

We   note also  that assumptions \eqref{e3.2}--\eqref{e3.4} do not preclude multi\-valued mappings $a$.  Such an example is
$$\begin{array}{rcl}
j(t,z)&\equiv&|z|(\log(|z|+1)),\vspace*{1,5mm}\\
a(t,z)&=&\(\log(|z|+1)+\displaystyle\frac1{|z|+1}\){\rm sign}\ z,\ \ \forall  z\in{\mathbb{R}}^d.\end{array}$$

By \eqref{e2.1}, one reduces equation \eqref{e3.1} to the random parabolic differential equation
\begin{equation}\label{e3.5}
\begin{array}{l}
\displaystyle\frac{{\partial} y}{{\partial} t}-e^{-W}{\rm div}_\xi a(t,\nabla(e^W(y+x))+({\lambda}+\mu)(y+x))=0\\
\hfill\mbox{ in }(0,T)\times\mathcal{O},\vspace*{1,5mm}\\
y=0\mbox{ on }(0,T)\times{\partial}\mathcal{O},\vspace*{1,5mm}\\
y(0,\xi)=0,\ \ \xi\in\mathcal{O}.\end{array}\end{equation}We are under the conditions of Section 2, where
$$\begin{array}{rcl}
 H&=&L^2(\mathcal{O}), \vspace*{1,5mm}\\
 A(t)y&=&-{\rm div}_\xi a(t,\nabla y),\vspace*{1,5mm}\\
  D(A(t))&=&\{y\in W^{1,1}_0(\mathcal{O});\ {\rm div}_\xi a(t,\nabla y)\in L^2(\mathcal{O})\}\vspace*{1,5mm}\\
  {\varphi}(t,y)&=&\displaystyle\int_\mathcal{O} j(t,\nabla y(\xi))d\xi.\end{array}$$

By \eqref{e2.11a}, we have
\begin{equation}\label{e3.6}
\psi^*(t,v)=\int_\mathcal{O}(a(t,\nabla z)\cdot\nabla z-j(t,\nabla z)+\frac\delta2\, z^2)d\xi,\ \forall  v\in L^2(\mathcal{O}),\end{equation}where $z$ is the solution to the equation
\begin{equation}\label{e3.7}
\begin{array}{rcll}
-{\rm div}\ a(t,\nabla z)+ \delta z&=&v&\mbox{ in }\mathcal{O},\vspace*{1,5mm}\\
z&=&0&\mbox{ on }{\partial}\mathcal{O},\end{array}
\end{equation}
or, equivalently,
\begin{equation}\label{e3-7prim}
z={\rm arg}\min_{\tilde z\in W^{1,1}_0(\mathcal{O})}
\left\{
\int_\mathcal{O} j(t,\nabla \tilde z)d\xi
-\int_\mathcal{O} v\tilde z\,d\xi+\frac\delta 2
\int_\mathcal{O} \tilde z^2d\xi\right\}.
\end{equation}
 By \eqref{e3.3}, it follows that \eqref{e3-7prim} has, for each $v\in L^2(\mathcal{O})$ and $t\in[0, T]$, a~unique solution $z\in W^{1,1}_0(\mathcal{O})$. In fact, as easily seen, by condition \eqref{e3.3} it follows that the functional arising in the right side part of \eqref{e3-7prim} is convex, lower semicontinuous and coercive on $W^{1,1}_0(\mathcal{O})$. By  \eqref{e2.23-z}), we have
\begin{equation}\label{e3.8}
 \begin{array}{ll}
\widetilde G_1(y,y_1)=\!\!\!&\displaystyle
{\mathbb{E}} \displaystyle\int^T_0   \int_\mathcal{O} (j(t,\nabla(e^{W(t)}(y(t)+x)))
+\displaystyle\frac\delta2\,|e^{W(t)}(y(t)+x)|^2_H\vspace*{1,5mm}\\
&+\,\displaystyle e^{W(t)}
((\nu+\delta)(y(t)+x)+\mu x)e^{W(t)}x)\,d\xi\,dt\vspace*{1,5mm}\\
&-{\mathbb{E}}\displaystyle\int^T_0(e^{W(t)}
y(t),e^{W(t)}(1+\mu)x)dt\vspace*{1,5mm}\\
&+ \displaystyle{\mathbb{E}}\int^T_0 \eta(e^{W(t)}y(t))dt
+ \displaystyle\frac12\,{\mathbb{E}} \int_\mathcal{O} |e^{W(T)}y_1(\xi)|^2 d\xi\vspace*{1,5mm}\\
&+{\mathbb{E}}(e^{W(T)}y_1,e^{W(T)}x),
\,(y,y_1)\in\mathcal{H}\times L^2({\Omega};H),\end{array}\hspace*{-6mm}
\end{equation}where (see \eqref{e2.16})
\begin{equation}\label{e3.9}
\eta(z)=(\nu+\delta)\int_\mathcal{O}|z|^2d\xi-\frac12\sum^{\infty}_{j=1}\mu^2_j\int_\mathcal{O}|z e_j|^2d\xi.\end{equation}
By \eqref{e2.18} and \eqref{e3.6}--\eqref{e3.7}, we also have \begin{equation}\label{e3.10}
\begin{array}{ll}
G_2(u)=\!\!\!
&\displaystyle{\mathbb{E}}\int^T_0   \int_\mathcal{O}(a(t,\nabla z(t,\xi))\cdot\nabla z(t,\xi))\\
&-\displaystyle j(t,\nabla z(t,\xi))+\displaystyle\frac\delta 2\, z^2(t,\xi)d\xi\, dt,\ u\in\mathcal{H},\end{array}\end{equation}
where $z(t,{\omega})\in  W^{1,1}_0(\mathcal{O} )$ for $dt\otimes\mathbb{P}$-a.e.,   $(t,{\omega})\in (0,T)\times{\Omega} $, is given by (see~\eqref{e3.7})
\begin{equation}\label{e3.11}
\begin{array}{rcll}-{\rm div}\ a(t,\nabla z)+ \delta z&=&u&\mbox{ in }\mathcal{O},\vspace*{1,5mm}\\
z&=&0&\mbox{ on }{\partial}\mathcal{O}.\end{array}\end{equation}
Taking into account that $a(t,\nabla z)\cdot\nabla z\ge j(t,\nabla z)-j(t,0)$, we see by \eqref{e3.3} and \eqref{e3.11} that
$$z\in L^1((0,T)\times{\Omega};W^{1,1}(\mathcal{O}))\cap L^2((0,T)\times\mathcal{O}\times{\Omega}).$$
Recalling \eqref{e1.7}--\eqref{e1.8}, we have
$$a(t,\nabla z)\cdot\nabla z-j(t,\nabla z)=j^*(t,a(t,\nabla z))\mbox{ a.e. in }(0,T)\times\mathcal{O},$$and this yields
\begin{equation}\label{e3.12}
\begin{array}{ll}
G_2(u)= \displaystyle{\mathbb{E}}\int^T_0
\int_\mathcal{O} (j^*(t,a(t,\nabla z(t,\xi)))
 +\displaystyle\frac\delta2\, z^2(t,\xi) )d\xi\,dt.\end{array}\end{equation}
By \eqref{e3.4}, it follows via the Dunford--Pettis weak compactness theorem in $L^1$ that every level set
$$\left\{v;{\mathbb{E}}\int^T_0\int_\mathcal{O} j^*(t,v(t,\xi))dx\,d\xi\le M\right\},\ M>0,$$is weakly compact in the space $
L^1((0,T)\times\mathcal{O}\times{\Omega}). $ By \eqref{e3.11} and \eqref{e3.12}, we see that, if
 $G_2(u_n)\le M,$ where $ \{u_n\}\subset L^2((0,T)\times\mathcal{O}\times{\Omega}\}$ and $z_n$ is the solution to \eqref{e3.11} with $u_n$ replacing $u$, then, by the Dunford--Pettis  theorem, the sequence $\{a(t,\nabla z_n)\}$ is weakly compact in $L^1 ((0,T)\times\mathcal{O}\times{\Omega}\}$. Hence $\{u_n\}$ is weakly compact in $L^1 ((0,T)\times{\Omega};W^{-1,{\infty}}(\mathcal{O}))$.

 By \eqref{e3.12}, it follows also that $\{z_n\}$ is weakly compact in $L^2((0,T)\times\mathcal{O}\times{\Omega}).$\break
    By \eqref{e2.22}, this means that, if $x\!\in\! L^2(\mathcal{O})$, then, for $(y,y_1,u)\!\in \!\mathcal{H}{\times} L^2({\Omega};H){\times}\mathcal{X}$,
\begin{equation}\label{e3.14}
\begin{array}{ll}
\overline G (y,y_1,u)\vspace*{1,5mm}\\
= \widetilde G_1(y,y_1)+
 \displaystyle{\mathbb{E}}\int^T_0\!\!\! \int_\mathcal{O}\!\bigg(j^*(t,a(t,\nabla z(t,\xi)) +\frac\delta2\, z^2(t,\xi))\!\bigg)d\xi\,dt, \end{array} \end{equation}
where   $z\in L^1((0,T)\times{\Omega};W^{1,-1}_0(\mathcal{O}))\cap L^2((0,T)\times\mathcal{O}\times{\Omega})$ is the solution to \eqref{e3.11}.

Let $(y_n,u_n)\in\mathcal{H}\times\mathcal{H}$ be such that $e^W{\mathcal{B}} y_n+u_n=0$ and $(y_n,u_n)\to(y,u)$ in $\mathcal{H}\times\mathcal{X},$ $y_n(T)\to y_1)$ in $L^2({\Omega};H)$.
Since $\sup\limits_n\{G_1(y_n)\}<{\infty}$, by \eqref{e3.3} and \eqref{e3.8}, it follows also that $\{\nabla(e^W(y_n+x)\}$ is weakly compact in $L^1 ((0,T)\times\mathcal{O}\times{\Omega})$, and so   $e^W(y+x)\in L^1 ((0,T)\times\mathcal{O};W^{1,1}_0(\mathcal{O})).$ Moreover,   it follows that $\left\{\frac{dy_n}{dt}\right\}$ is weakly compact in $L^1((0,T)\times{\Omega};W^{-1,{\infty}}(\mathcal{O}))$, and so $\frac{dy}{dt}\in L^1((0,T)\times{\Omega};W^{-1,{\infty}}(\mathcal{O})).
$ This implies that the equation  $\widetilde{\mathcal{B}}(y^*,y^*_1)+u^*=0$ reduces to
$$\begin{array}{l}
e^W\,\displaystyle\frac{dy^*}{dt}+e^W
(\mu+\nu+\delta)(y^*+x)
+u^*=0\mbox{ in }{\mathcal{D}}'(0,T),\ \mathbb{P}\mbox{-a.s.},\vspace*{1,5mm}\\
y^*(0)=0,\quad y^*(T)=y^*_1.\end{array}$$
Hence, if $D(\overline G_1)=\{(y,y_1,u);\,\overline G_1(y,y_1,u)<{\infty}\}$, then we have
$$\begin{array}{l}
D(\overline G_1)\subset\{(y,y_1,u)\in\mathcal{H}\times L^2({\Omega};H)\times\mathcal{H};\,e^Wy\in L^1((0,T)\times{\Omega}; W^{1,1}_0(\mathcal{O}));\vspace*{1,5mm}\\
\displaystyle\frac{dy}{dt}\in L^1((0,T)\times{\Omega};W^{-1,{\infty}}(\mathcal{O}));\,u\in L^1((0,T)\times{\Omega};W^{-1,{\infty}}(\mathcal{O})),
\,y_1=y(T)\}.\end{array}$$
This means that, in this case, problem (P) can be rewritten as
\begin{equation}\label{e3.15}
\begin{array}{ll}
{\rm Min}\Big\{\!\!\!&\overline G(y,y(T),u);\,y\in L^2((0,T)\times\mathcal{O}\times{\Omega})\cap \mathcal{H},\\
& e^W(y+x)\in L^1((0,T)\times{\Omega};W^{1,1}_0(\mathcal{O})),\vspace*{1,5mm}\\
&\displaystyle\frac{dy}{dt}\in L^1((0,T)\times{\Omega};W^{-1,{\infty}}(\mathcal{O})),\vspace*{1,5mm}\\  & u\in L^1((0,T)\times{\Omega};W^{-1{\infty}}(\mathcal{O}))\cap\mathcal{X};\vspace*{1,5mm}\\
&\mbox{subject to }\vspace*{1,5mm}\\
&\displaystyle\frac{dy}{dt}+(\mu+\nu+\delta)(y+x)+e^{-W} u=0\mbox{ on }(0,T);\,y(0)=0\Big\},\end{array}
\hspace*{-10mm}\end{equation}
where $\overline G_1$ is defined by \eqref{e3.14}.
By Theorem \ref{t2.1}, there is a unique solution $(y^*,u^*)$ to \eqref{e3.15}. Taking into account that $ u^*\in L^1((0,T)\times{\Omega}; W^{-1,{\infty}}(\mathcal{O}))$ and that
$$y^*(t)=-\int^t_0 e^{-W}u^*(s)ds-\int^t_0(\mu+\nu+\delta)(y^*(s)+x)ds,\ \forall  t\in(0,T),$$ we infer that the process $t\to y^*(t)$ in pathwise $W^{-1,{\infty}}(\mathcal{O})$ continuous on$(0,T)$. By Theorem \ref{t2.4-b}, we have, therefore,

\begin{theorem}\label{t3.1} Assume that $x\in L^2(\mathcal{O})$ and that conditions \eqref{e3.2}--\eqref{e3.4} hold. Then, equation \eqref{e3.1} has a unique variational solution
\begin{equation}\label{e3.16}
X^*\in L^2((0,T)\times\mathcal{O}\times{\Omega}),\ e^WX^*\in L^1((0,T)\times{\Omega};W^{1,1}_0(\mathcal{O})).\end{equation}
Moreover, the process $t\to X^*(t)$ is $(\mathcal{F}_t)_{t\ge0}$-adapted and pathwise $W^{-1,{\infty}}(\mathcal{O})$-valued  continuous on $(0,T)$.\end{theorem}

\paragraph{The total variation flow}\ \bigskip

\noindent  The stochastic differential equation
\begin{equation}\label{e3.17}
\begin{array}{l}
dX-{\rm div}\(\displaystyle\frac{\nabla X}{|\nabla X|_d}\)dt+{\lambda} X\,dt=X\,dW\mbox{ in }(0,T)\times\mathcal{O},\vspace*{1,5mm}\\
X(0)=x\mbox{ in }\mathcal{O},\vspace*{1,5mm}\\
X=0\mbox{ on }(0,T)\times{\partial}\mathcal{O}\end{array}
\end{equation}
with $x\in L^2(\mathcal{O})$ is the equation of stochastic variational flow in $\mathcal{O}\subset{\mathbb{R}}^d$, $1\le d\le 3$. The existence and uniqueness of a generalized solution to \eqref{e3.17} $X:[0,T]\to BV(\mathcal{O})$ was established in \cite{5} by using some specific approximation techniques. We shall treat now equation \eqref{e3.17} in the framework of variational solution developed above in the space $H=L^2(\mathcal{O})$ with the norm $|\cdot|_H=|\cdot|_2$ and the scalar product $(\cdot,\cdot)$, and ${\varphi}:L^2(\mathcal{O})\to\overline{\mathbb{R}}$ defined by
$${\varphi}(y)=\left\{\begin{array}{ll}
\|Dy\|+\displaystyle\int_{{\partial}\mathcal{O}}|{\gamma}_0(y)|d\mathcal{H}^{d-1},&y\in BV(\mathcal{O})\setminus L^2(\mathcal{O}),\vspace*{1,5mm}\\
+{\infty}&\mbox{otherwise}.\end{array}\right.$$Here, $BV(\mathcal{O})$ is the space of functions with bounded variation and $\|Dy\|$ is the total variation of $y\in BV(\mathcal{O}).$ (See, e.g., \cite{5}.)
Then, with the notations of Section 2, we have  $Ay={\partial}{\varphi}(y)$, where ${\partial}{\varphi}:L^2(\mathcal{O})\to L^2(\mathcal{O})$ is the subdifferential of ${\varphi}$ and (see \eqref{e2.2}, \eqref{e2.17})
\begin{equation}\label{e3.18}
\begin{array}{l}
\displaystyle\frac{{\partial} y}{{\partial} t}+e^{-W}A(e^W(y+x))+\mu(y+x)=0\mbox{ in }(0,T)\times\mathcal{O},\vspace*{1,5mm}\\
y(0,\xi)=0,\ \ \xi\in\mathcal{O},\vspace*{1,5mm}\\
y=0\mbox{ on }(0,T)\times{\partial}\mathcal{O}.\end{array}\end{equation}
The function $\widetilde G_1$ is given, in this case, by

\begin{equation}\label{e3.19}
\begin{array}{ll}
\widetilde G_1(y,y_1)=\!\!\!&{\mathbb{E}}\displaystyle\int^T_0 \bigg({\varphi}(e^{W(t)}(y(t)+x))+
\displaystyle\frac\delta2\,|e^{W(t)}(y(t)+x)|^2_2\vspace*{1,5mm}\\
&+\displaystyle (e^{W(t)}((\nu+\delta)(y(t)+x)+
\mu  x),e^{W(t)}x)\bigg)dt  \vspace*{1,5mm}\\
&-{\mathbb{E}}\displaystyle\int^T_0(e^{W(t)}y(t),e^{W(t)}(1+\mu)x)dt\vspace*{1,5mm}\\
&+ {\mathbb{E}}\displaystyle\int^T_0 \eta(e^{W(t)}y(t))dt+
\displaystyle\frac12\,{\mathbb{E}}|e^{W(T)}y_1|^2_2\vspace*{1,5mm}\\  &+{\mathbb{E}}(e^{W(T)}y_1,e^{W(T)}x),\,
(y,y_1)\in\mathcal{H}\times L^2({\Omega};H),\end{array}\end{equation}where $\eta$  is given by \eqref{e3.9}.
We have also (see \eqref{e2.11}, \eqref{e2.11a}, \eqref{e3.6})
$$\begin{array}{rcll}
\psi(y)&=&{\varphi}(y)+\displaystyle\frac\delta2\,|y|^2_2,& \forall  y\in D({\varphi}),\vspace*{1,5mm}\\
\psi^*(v)&=&(v,\theta)-{\varphi}(\theta)
-\displaystyle\frac\delta2\,|\theta|^2_2,
&v\in{\partial}{\varphi}(\theta)+\delta \theta.\end{array}$$
Hence,
$$\begin{array}{lcl}
\psi^*(v)&=&\displaystyle\frac\delta2\,|\theta|^2
+({\partial}{\varphi}(z),\theta)-{\varphi}(\theta)\vspace*{1,5mm}\\
&=&\displaystyle\frac\delta2\,|(\delta I+{\partial}{\varphi})^{-1}v|^2_2+{\varphi}^*(v-(\delta I+{\partial}{\varphi})^{-1}v)\end{array}$$
and, therefore, by \eqref{e2.18},
$$G_2(u)={\mathbb{E}}\int^T_0
\(\displaystyle\frac\delta2\,|(\delta I+{\partial}{\varphi})^{-1}(u)|^2_2
+{\varphi}^*(u-(\delta  I+{\partial}{\varphi})^{-1}( u))\)dt,$$where ${\varphi}^*:L^2(\mathcal{O})\to\overline{\mathbb{R}}$ is the conjugate of the function ${\varphi}$.    This yields
\begin{equation}\label{e3.20}
\begin{array}{ll}
\overline G(y,y_1,u)=\!\!\!
&G_1(y,y_1)+
\displaystyle\liminf_{^{(z,v)\to (y,u)}_{\  {\rm in\ }  \mathcal{H}\times\mathcal{X}}}
\Big\{{\mathbb{E}}\int^T_0\Big(
\displaystyle\frac\delta2\,
|(I+{\partial}{\varphi})^{-1}(v(t))|^2_2\vspace*{1,5mm}\\
&\hfill\displaystyle+{\varphi}^*(v-( \delta I+{\partial}{\varphi})^{-1}(v(t)))\Big)dt
,\,e^W{\mathcal{B}} z+v=0\Big\}, \end{array}\end{equation}
where the space $\mathcal{X}$ is defined by \eqref{e2.20a}.

By definition, the solution $(y^*,y^*_1)$ to the minimization problem
\begin{equation}\label{e3.21}
{\rm Min}\{\overline G(y,y_1,u) ;\ {\mathcal{B}} (y,y_1)+u=0,\ (y,y_1,u)\in\mathcal{H}\times L^2({\Omega};H)\times\mathcal{X}\}\end{equation}  is the {\it variational solution} to the random differential equation \eqref{e3.18}.
\newpage

Denote by $V^*$ the dual of the space $V=BV(\mathcal{O})\cap L^2(\mathcal{O})$. We note that ${\varphi}^*$ can be extended as a convex lower semicontinuous convex function on $F^*$, and we also have
$$\frac{{\varphi}^*(u)}{\|u\|_{V^*}}
\longrightarrow
+{\infty}\mbox{\ \ as\ \ }\|u\|_{V^*}\longrightarrow +{\infty}.$$
Then, if $(z_n,y_n)\in\mathcal{H}\times\mathcal{H}$ is convergent to $(y,u)\in\mathcal{H}\times\mathcal{X}$, it follows by the Dunford-Pettis compactness criterium (see \cite{11-a}) that $\{v_n\}$ is weakly compact in $L^1((0,T)\times{\Omega};V^*)$. This implies that
$$D(\overline G)\subset L^1((0,T)\times{\Omega};BV(\mathcal{O}))\times L^2({\Omega};H)\times L^1((0,T)\times{\Omega};V^*),$$and so, in particular, it follows that
$$y\in W^{1,1}([0,T];V^*),\ \mathbb{P}\mbox{-a.s.}$$We have, therefore,
\begin{theorem}\label{t3.2} Let $x\in BV(\mathcal{O})\cap L^2(\mathcal{O})$. Then equation \eqref{e3.17} has a unique varia\-tio\-nal solution $X=e^W(y+x)$ which is $V^*$-valued pathwise continuous and satisfies
\begin{eqnarray}
 {\varphi}(X)&\in & L^1((0,T)\times{\Omega}),\label{e3.22}\\[1mm]
 X&\in & L^2((0,T)\times\mathcal{O}\times{\Omega}),\ AX\in L^1((0,T)\times{\Omega};V^*),\label{e3.23}\\[1mm]
 e^{-W}X&\in&W^{1,1}([0,T];V^*),\ \mathbb{P}\mbox{-a.s.}\label{e3.23-az}
\end{eqnarray}
\end{theorem}

In \cite{5}, it was proved the existence and uniqueness of a generalized solution $X$, also called {\it the variational solution}, which was obtained as limit $X^*=\lim\limits_{{\varepsilon}\to0}X_{\varepsilon}$ in $L^2({\Omega};C((0,T);L^2(\mathcal{O})))$, where $X_{\varepsilon}$ is the solution to the approximating equation
\begin{equation}\label{e3.24}
\begin{array}{l}
dX_{\varepsilon}-{\rm div}\ a_{\varepsilon}(\nabla X_{\varepsilon})dt+{\lambda} X_{\varepsilon}=X_{\varepsilon} dW\mbox{ in }(0,T)\times\mathcal{O},\vspace*{1,5mm}\\
X_{\varepsilon}(0)=x,\ \ \ X_{\varepsilon}=0\mbox{ on }(0,T)\times\mathcal{O},\end{array}\end{equation}where $a_{\varepsilon}=\nabla j_{\varepsilon}$ and $j_{\varepsilon}$ is the Moreau--Yosida approximation of the function $r\to|r|_d$. Since, as strong solution to \eqref{e3.24}, $X_{\varepsilon}$ is also a variational solution to this equation in sense of Definition \ref{d2.2}, it is clear by the structural stability of convex minimization problems that, for ${\varepsilon}\to0$, we have also $X_{\varepsilon}\to X$, where $X$ is the variational solution given by Theorem \ref{t3.2}.

We may infer, therefore, that {\it the function $X$ given by Theorem {\rm\ref{t3.2}} is just the generalized solution of \eqref{e3.17} given by Theorem  {\rm3.1}} in \cite{5}. In particular, this implies that $X$ is $L^2(\mathcal{O})$-valued pathwise continuous.

In \cite{3-a}, it is de\-ve\-loped a direct variational approach to \eqref{e3.17}, which leads via first order conditions of optimality  to sharper results. (On these lines, see also \cite{10-a}.)

\paragraph{Stochastic porous media equations}\ \bigskip

\noindent  Consider the equation
\begin{equation}\label{e3.25}
\begin{array}{l}
dX-\Delta\beta(X)dt+{\lambda} X\,dt=X\,dW\mbox{ in }(0,T)\times\mathcal{O},\vspace*{1,5mm}\\
X=0\mbox{ on }(0,T)\times{\partial}\mathcal{O},\vspace*{1,5mm}\\
X(0,\xi)=x(\xi),\ \ \xi\in\mathcal{O},\end{array}
\end{equation}
where $\mathcal{O}$ is a bounded and open domain of ${\mathbb{R}}^d$, $d\ge1$, ${\lambda}>0$, $W$ is a Wiener process in $H=H^{-1}(\mathcal{O})$ of the form \eqref{e1.2} and ${\beta}$ is a continuous and monotonically nondecreasing function such that $\beta(0)=0$ and
\begin{equation}\label{e3.26}
\lim_{|r|\to{\infty}}\ \frac{j(r)}{|r|}=+{\infty} .
\end{equation}
In this case,
$$\begin{array}{rcl}
H&=&H^{-1}(\mathcal{O}) ,\vspace*{1,5mm}\\  Ay&=&-\Delta\beta(y) ,\vspace*{1,5mm}\\
D(A)&=&\{y\in H^{-1}(\mathcal{O})\cap L^1(\mathcal{O}),\
\beta(y)\in H^1_0(\mathcal{O})\}\mbox{ and }\vspace*{1,5mm}\\
  A&=&{\partial}{\varphi},\mbox{ where ${\varphi}(y)=\displaystyle\int_\mathcal{O} j(y(\xi))d\xi.$}\end{array}$$
By \eqref{e2.11}, we have also
$$\psi^*(v)=\int_\mathcal{O} j^*(\beta(\theta))d\xi
+\frac\delta2\,|\theta|^2_{-1},\ v\in L^2(\mathcal{O}),$$where $\theta\in H^{-1}(\mathcal{O})\cap L^1(\mathcal{O}),$
$$\begin{array}{rcll}
2\delta \theta-\Delta\beta(\theta)&=&v&\mbox{ in }\mathcal{O},\vspace*{1,5mm}\\
\theta&=&0&\mbox{ on }{\partial}\mathcal{O},\end{array}$$and $|\cdot|_{-1}$ is the norm of $H^{-1}(\mathcal{O})$. Then we have
$$\begin{array}{ll}
\widetilde G_1(y,y_1)=\!\!\!&\displaystyle{\mathbb{E}}\int^T_0 \(\int_\mathcal{O} j(e^{W(t)}(y(t)+x))d\xi+
\displaystyle\frac\delta2\,|e^{W(t)}(y(t)
+x)|^2_{-1}\)d\xi\,dt\vspace*{1,5mm}\\
&+\,\displaystyle{\mathbb{E}}\int^T_0 \int_\mathcal{O} e^{W(t)}((\nu+\delta)
(y(t)+x)+\mu  x)e^{W(t)}x\,d\xi\,dt\vspace*{1,5mm}\\
&-{\mathbb{E}}\displaystyle\int^T_0(e^{W(t)}y(t),e^{W(t)}(1+\mu)x)dt\vspace*{1,5mm}\\
&\displaystyle+\,{\mathbb{E}}\int^T_0 \eta(e^{W}y)dt
+\displaystyle\frac12\,{\mathbb{E}}|e^{W(T)}y(T)|^2_{-1}\vspace*{1,5mm}\\
&+(e^{W(T)}y_1,e^{W(T)}x)_{-1},\end{array}$$
while
$$G_2(u)=
 \displaystyle{\mathbb{E}}\int^T_0 \Bigg(\int_\mathcal{O} j^*(\beta(\widetilde z))d\xi+\frac\delta2\,|\widetilde z(t)|^2_{-1}\Bigg)dt ,$$where
\begin{equation}\label{e3.27}
\begin{array}{rcll}
 \delta\widetilde z-\Delta\beta(\widetilde z)&=&u&\mbox{ in }\mathcal{O},\vspace*{1,5mm}\\
\widetilde z&=&0&\mbox{ on }{\partial}\mathcal{O}.\end{array}
\end{equation}
(Here, $(\cdot,\cdot)_{-1}$ is the scalar product of $H^{-1}(\mathcal{O}).$)

Taking into account that $\frac{j^*(r)}{|r|}\to+{\infty}$ as $|r|\to{\infty}$, it follows, as in the previous case, for each $M>0$, the set
$$\left\{\beta(\widetilde z);\ {\mathbb{E}}\int^T_0\int_\mathcal{O} j^*(\beta(\widetilde z))dt\,d\xi\le M\right\}$$is weakly compact in $L^1((0,T)\times\mathcal{O}\times{\Omega})$, we infer that
\begin{equation}\label{e3.28}
\overline G(y,y_1,u)=\widetilde G_1(y,y_1)+ {\mathbb{E}}\displaystyle\int^T_0 \bigg(\int_\mathcal{O} j^*(\beta(\widetilde z))d\xi
+\frac\delta2\,|\widetilde z(t)|^2_{-1}\bigg)dt,
 \end{equation}
where $\widetilde z$ is the solution to \eqref{e3.27}. This implies that
$$D(\overline G)\subset\{(y,y_1,u)\in\mathcal{H}\times L^2({\Omega};H)\times\mathcal{X};\,u\in L^1((0,T)\times{\Omega};{\mathcal{Z}})\}.$$
Here ${\mathcal{Z}}=(-\Delta)^{-1}(L^1(\mathcal{O}))\subset W^{1,p}_0(\mathcal{O}),\ 1\le p<\frac d{d-1}$, where $\Delta$ is the Laplace operator with homogeneous Dirichlet conditions and $$D(\overline G)=\{(y,y_1,u);\ \overline G(y,y_1,u)<{\infty}\}.$$

We define, as above, the solution to \eqref{e3.25} as $X^*=e^Wy^*$, where $(y^*,y^*_1,u^*)$ is the solution to the minimization problem
\begin{equation}\label{e3.29}
\begin{array}{r}
{\rm Min} \bigg\{\overline G(y,y_1,u); \displaystyle \frac{dy}{dt}{+}
(\mu{+}\nu{+}\delta)(y{+}x){+}e^{-W}u{=}0,\, y(0)=0,\vspace*{1,5mm}\\  y(T){=}y_1,\
(y,y_1,u)\in\mathcal{H}\times L^2({\Omega};H)\times\mathcal{X}
\bigg\}\!\!.\end{array}
\end{equation}
(Here, $\frac{dy}{dt}$ is taken in sense of distributions, i.e., in ${\mathcal{D}}'(0,T;H)$.) We have, therefore,
\begin{theorem}\label{t3.2a} Assume that $x\in L^2(\mathcal{O})$. Then equation \eqref{e3.25} has a unique variational solution $X^*$,
$$\begin{array}{c}
X^*\in L^2((0,T)\times\mathcal{O}\times{\Omega});\,{\varphi}(X^*)
\in L^1((0,T)\times\mathcal{O}\times{\Omega}),\vspace*{1,5mm}\\
e^{-W}X\in W^{1,1}([0,T];W^{1,1}_0(\mathcal{O})),\ \mathbb{P}\mbox{-a.s.}\end{array}$$Moreover, the process $t\to X^*(t)$ is pathwise $W^{1,1}_0(\mathcal{O})$-valued continuous on~$(0,T).$
\end{theorem}

\begin{remark}\rm A different treatment of equation \eqref{e3.25} under the general assumptions \eqref{e3.26} was developed in \cite{4a} (see also \cite{4aa}, Ch. 5).\end{remark}

\section{Stochastic variational inequalities}
\setcounter{equation}{0}

Consider the stochastic differential equation
\begin{equation}\label{e4.1}
\begin{array}{l}
dX+A_0X\,dt+N_K(X)dt+{\lambda} X\,dt\ni X\,dW,\ t\in(0, T),\vspace*{1,5mm}\\
X(0)=x,\end{array}\end{equation}in a real Hilbert space $H$ with the scalar product $(\cdot,\cdot)$ and the norm $|\cdot|$. Assume that $x\in H$ and
\begin{itemize}
\item[(j)] {\it $A_0:D(A_0)\subset H\to H$ is a linear self-adjoint, positive definite operator in $H$.}
\item[(jj)] {\it $W$ is the Wiener process \eqref{e1.2} and ${\lambda}>\nu$.}
\item[(jjj)] {\it $K$ is a closed, convex subset of $H$ such that $0\in K$, $(I+{\lambda} A_0)^{-1}K\subset K$, $\forall  {\lambda}>0.$ } \end{itemize}
    Here, $N_K:H\to 2^H$ is the normal cone to $K$, that is,
\begin{equation}\label{e4.2}
N_K(u)=\{\eta\in H;\ (\eta,u-v)\ge0,\ \forall  v\in K\}.\end{equation}

By the transformation \eqref{e2.1}, equation \eqref{e4.1} reduces to the nonlinear random differential equation
\begin{equation}\label{e4.3}
\begin{array}{l}
\displaystyle\frac{dy}{dt}+e^{-W}A_0(e^W(y+x))+e^{-W}N_K(e^W(y+x))+\mu(y+x)=0,\\\hfill t\in(0,T),\vspace*{1,5mm}\\
y(0)=0.\end{array}\end{equation}
(We note that, if $W(t)=\sum\limits^N_{j=1}\mu_j{\beta}_j(t)$, then \eqref{e4.3} reduces to a deterministic variational inequality.)

To represent this problem as an optimization problem of the form (P), we~set
$${\varphi}(u)=\frac12\ (A_0u,u)+I_K(u),\ \forall  u\in H,$$where $I_K$ is the indicator  function $$I_K(u)=\left\{\begin{array}{ll}
0&\mbox{ if }u\in K,\\
+{\infty}&\mbox{ otherwise}.\end{array}\right.$$
The function ${\varphi}:H\to]-{\infty},+{\infty}]$ is convex  and lower semicontinuous. Then, by \eqref{e2.6}, \eqref{e2.17}, \eqref{e2.18}, we have
\begin{eqnarray}
\widetilde G_1(y,y_1)&=&{\mathbb{E}}\int^T_0 \bigg(\frac12\,(A_0(e^{W(t)}(y(t)+x)),e^{W(t)}(y(t)+x))\ \ \ \label{e4.4}\\
&&+\,e^{2W}((\nu+\delta)(y+x)+\mu y)x
+\displaystyle\frac\delta2\,|e^{W(t)}(y(t)+x)|^2_H\nonumber\\ &&+I_K(e^{W(t)}(y(t)+x))
+\eta(e^{W(t)}y(t))\bigg)dt\nonumber\\
&&-{\mathbb{E}}\displaystyle\int^T_0(e^Wy,e^W(1+\mu)y)dt\vspace*{1,5mm}\\
&&+\displaystyle\frac12\,{\mathbb{E}}|e^{W(T)}y_1|^2_H
+{\mathbb{E}}(e^{W(T)}y_1,e^{W(T)}x),\nonumber\\
G_2(u)&=&{\mathbb{E}}\int^T_0  \psi^*(u(t) )dt
 , \label{e4.5}
\end{eqnarray}where, by \eqref{e2.11}-\eqref{e2.11a}, we have
\newpage
\begin{equation}\label{e4.5a}
\begin{array}{ll}
\psi^*(e^Wu)\!\!\!
&=\displaystyle\sup\Bigg\{(e^Wu,v)-\frac12\,(A_0v,v)-
\frac\delta2\,|v|^2;\,v\in K\Bigg\}\\
&=(e^Wu,z)-\displaystyle\frac12\,(A_0z,z)-\displaystyle\frac\delta2|z|^2,\end{array},
\end{equation}where $A_0z+\delta z+N_K(z)\ni e^Wu.$ (We note that, by (iii), $z$ is uniquely defined.)

By \eqref{e4.4}-\eqref{e4.5}, we see that
$$\overline G(y,y_1,u)=\widetilde G_1(y,y_1)+\frac12\,\liminf_{n\to{\infty}}
{\mathbb{E}}\int^T_0\((A_0z_n,z_n)+\frac\delta2\,|z_n|^2\)dt,$$
where
\begin{equation}\label{e4.6-a}
\begin{array}{l}
A_0z_n+\delta z_n+N_K(z_n)\ni  u_n,\ e^W By_n+u_n=0,\vspace*{1,5mm}\\
 y_n\to y\mbox{ in }\mathcal{H},\ y_n(T)\to y_1\mbox{ in }L^2({\Omega};H),\ u_n\to u\in\mathcal{X}.\end{array}\end{equation}This  yields
\begin{equation}\label{e4.7-b}
{\mathbb{E}}\int^T_0|A^{\frac12}_0z_n|^2 dt\le C<{\infty},\ \forall  n\in{\mathbb{N}},\end{equation}
and, therefore, we have
\begin{equation}\label{e4.9}
\begin{array}{rl}
\overline G(y,y_1,u)\!\!\!
&=\widetilde G_1(y,y_1)+\displaystyle\frac12\,{\mathbb{E}}\int^T_0
(|A^{\frac12}_0z|^2+\delta|z|^2)dt,\vspace*{1,5mm}\\
z\!\!\!&=w-\displaystyle\lim_{n\to{\infty}}z_n\mbox{ in }L^2((0,T)\times{\Omega};V),\end{array}
\end{equation}where $V=D(A^{\frac12}_0).$
We note that $D(\widetilde G_1)\subset L^2((0,T)\times{\Omega};V).$

We may conclude, therefore, by Theorem \ref{t2.4-b} that

\begin{theorem}\label{t4.1-b} Under hypotheses {\rm(j)--(jjj)}, there is a unique variational solution $X^*(t)\in K$, a.e. $t\in(0,T)$, $X^*\in L^2((0,T)\times V;{\Omega})$ to equation \eqref{e4.1}.
\end{theorem}

More insight into the problem can be gained in the following two special cases.

\paragraph{Stochastic parabolic variational inequalities} \ \smallskip

\noindent  The stochastic differential equation
\begin{equation}\label{e4.18}
\begin{array}{l}
dX-\Delta X\,dt+{\lambda} X\,dt+N_K(X)dt\ni X\,dW\mbox{ in }(0,T)\times\mathcal{O},\vspace*{1,5mm}\\
X(0)=x\mbox{ in }\mathcal{O},\vspace*{1,5mm}\\
X=0\mbox{ on }(0,T)\times{\partial}\mathcal{O},\end{array}\end{equation}
 where $N_K(X){\subset} L^2(\mathcal{O})$ is the normal cone to the closed convex set $K$ of $L^2(\mathcal{O})$,
 \newpage
$$K=\{z\in L^2(\mathcal{O});\ z\ge0,\mbox{ a.e. in }\mathcal{O}\},\ \alpha\in{\mathbb{R}},$$can be   treated following the above infinite-dimensional scheme in the space $H=L^2(\mathcal{O})$, where $A_0u=-\Delta u,$ $u\in D(A_0)=H^1_0(\mathcal{O})\cap H^2(\mathcal{O})$.

Then the variational solution to \eqref{e4.18} is defined  by $X=e^Wy$, where $y$ is given by \eqref{e4.8-b} and $\overline G$ is given by
$$\overline G(y,y_1,u)=\widetilde G_1(y,y_1)+\frac 12\,{\mathbb{E}}\int^T_0\int_\mathcal{O}(|\nabla z^2|^2+\delta|z|^2)d\xi\,dt,$$
where $G_1$ is defined y \eqref{e4.4} and $z=w-\lim\limits_{n\to{\infty}}z_n$ in $L^2((0,T)\times{\Omega};H^1_0(\mathcal{O}))$,
\begin{equation}\label{e4.11-b}
\begin{array}{c}
-{\Delta} z_n+\delta z_n+\eta_n=u_n,\ e^W {\mathcal{B}} y_n+y_n=0,\vspace*{1,5mm}\\
\eta_n\in N_K(z_n),\ {\mathbb{E}}\displaystyle\int^T_0\int_\mathcal{O}|\nabla z_n|^2d\xi\,dt\le C,\ \forall  n.\end{array}
\end{equation}
Since $u_n\to  u$ in ${\mathcal{D}}'(0,T;L^2(\mathcal{O}))$ and $\eta_n(t,\xi)\le0$ a.e. $(t,\xi)\in(0,T)\times\mathcal{O},$  by \eqref{e4.11-b}, we infer that
$$-\Delta z+\delta z+\eta= u\mbox{ in }{\mathcal{D}}'((0,T)\times\mathcal{O}),$$
where  $\eta,u $ are in $\mathcal{M}((0,T)\times\mathcal{O})$  the space of bounded measures on \mbox{$(0,T)\times\mathcal{O}$.} If we denote by $\eta_a,u_a\in L^1((0,T)\times\mathcal{O})$ the  absolutely con\-ti\-nuous parts of $\eta$ and $u$, we get
 $$\begin{array}{c}
 -\Delta z+\delta z+\eta_a= u_a\mbox{ in }L^1(\mathcal{O}),\vspace*{1,5mm}\\
 z\in H^1_0(\mathcal{O})\mbox{ and }\eta_a(t,\xi)=0,\mbox{ a.e.  on }[z(t,\xi)>0]\vspace*{1,5mm}\\
 \eta_a(t,\xi)\ge0\mbox{, a.e. on }[z(t,\xi)=0].\end{array}$$
Then the process $X=e^W(y+x)$ is the variational solution to \eqref{e4.18} and so, by Theorem \ref{t4.1-b}, we have

\begin{corollary}\label{c4.2}There is a unique variational solution $X{\in} L^2((0,T){\times}{\Omega};H^1_0(\mathcal{O})),$ $X\ge0$, a.e. on $(0,T)\times{\Omega}.$
\end{corollary}

 \paragraph{Finite   dimensional   stochastic variational inequalities}\ \smallskip

   \noindent  Consider equation \eqref{e4.1} in the special case $K\subset{\mathbb{R}}^d$, ${\rm int}\ K\ne\emptyset,$ $0\in K$, $W=\sum\limits^N_{i=1}\mu_i\beta_i $ and $A_0\in L({\mathbb{R}}^d,{\mathbb{R}}^d)$, $A_0=A^*_0$. Then, as easily seen by \eqref{e2.12}, we have
   \newpage
\begin{equation}\label{e4.9a}
\psi^*(u)\ge\alpha_1|u|-\alpha_2,\ \forall  u\in{\mathbb{R}}^d.\end{equation}
Let $z_n$ be the solution to (see \eqref{e4.6-a})

 \begin{equation}\label{e4.14}
A_0z_n+\delta z_n+N_K(z_n)\ni u_n.\end{equation}
Since, by \eqref{e4.9a}-\eqref{e4.14}, the sequence $\{ u_n\}$ is bounded in $L^1 ((0,T)\times{\Omega},{\mathbb{R}}^d)$, it follows that it is weak-star compact in $\mathcal{M}(0,T;{\mathbb{R}}^d)$, $\forall {\varepsilon}>0$, and so $u\in\mathcal{M}(0,T;{\mathbb{R}}^d)$. (Here, $\mathcal{M}(0,T;{\mathbb{R}}^d)$ is the space of ${\mathbb{R}}^d$-valued bounded measures on $(0,T)$. Letting $n\to{\infty}$ in \eqref{e4.14}, we get
\begin{equation}\label{e4.11-a}
A_0z+\delta z+\zeta= u,\end{equation}
where $ u\in \mathcal{M}(0,T;{\mathbb{R}}^d)$, $\forall  {\varepsilon}>0,$ and $\zeta\in\mathcal{M}((0,T);{\mathbb{R}}^d)$,  $\mathbb{P}\mbox{-a.s.}$ By the Lebesgue decomposition theorem, we have
$$\begin{array}{ll}
\zeta=\zeta_a+\zeta_s,&\zeta_a\in L^1 (0,T;{\mathbb{R}}^d),\vspace*{1,5mm}\\
u=u_a+u_s,&u_a\in L^1 (0,T;{\mathbb{R}}^d),\end{array}$$where $u_s$ and $\zeta_s$ are singular measures and $\zeta_a\in N_K(z).$  Hence, by \eqref{e4.11-a}, we  have
\begin{equation}\label{e4.15prim}
z=(A_0+\delta I+N_K)^{-1}( u_a)=F(u_a),\ \zeta_s= u_s.\end{equation}
As a matter of fact, the singular measure $\zeta_s$ belongs to the normal cone $N_{\mathcal{K}}(z)\subset\mathcal{M}(0,T;{\mathbb{R}}^d)$ to the set ${\mathcal{K}}=\{\widetilde z\in C([0,T];{\mathbb{R}}^d);$ $\widetilde z(t)\in K,$ \mbox{$\forall  t\in[0,T]\}$} and it is concentrated on the set of $t$-values for which $z(t)$  defined by \eqref{e4.15prim} lies on the boundary ${\partial} K$ of $K$.

By \eqref{e2.20a}-\eqref{e2.21}, we have
\begin{equation}\label{e4.15}
\overline G(y,y_1,u)
 =\widetilde G_1(y,y_1)+{\mathbb{E}}\displaystyle\int^T_0
 \Bigg(\frac12(A_0F(u_a),F(u_a))
 +\frac\delta2\,|F(u_a)|^2\Bigg)dt,
 \end{equation}where $\widetilde G_1$ is given by \eqref{e4.4} and $y\in\mathcal{H}$ is solution to the equation

\begin{equation}\label{e4.16}
\begin{array}{ll}
y=y_a+y_s,\ y_a\in AC([0,T];{\mathbb{R}}^d),\ y_s\in BV([0,T];{\mathbb{R}}^d),\ \mathbb{P}\mbox{-a.s.},\vspace*{1,5mm}\\
\displaystyle\frac{dy_a}{dt}+(\mu+\nu+\delta)
(y_a+x)+e^{-W}u_a
=0,\mbox{ a.e. on }(0,T),\vspace*{1,5mm}\\
y_a(0)=0,\vspace*{1,5mm}\\
\displaystyle\frac{dy_s}{dt}+ e^{-W}u_s=0\mbox{ in }{\mathcal{D}}'(0,T;{\mathbb{R}}^d),\end{array}\end{equation}
where $BV([0,T];{\mathbb{R}}^d)$ is the space of functions with  founded variations on $[0,T]$. We note that,  by \eqref{e4.15}, it follows also that
$$\begin{array}{r}
D(\overline G) \subset \{(y,y_1,u) \in \mathcal{H}\times L^2({\Omega};H)
 \times \mathcal{X};\ y \in  BV([0,T];{\mathbb{R}}^d),\  \mathbb{P}\mbox{-a.s.},\vspace*{1,5mm}\\  F(u_a) \in  L^2((0,T) \times {\Omega} \times {\mathbb{R}}^d)\},\end{array}$$where $D(\overline G)=\{(y,y_1,u);\ G(y,y_1,u)<{\infty}\}.$
We have, therefore,

\begin{theorem}\label{t4.1} The minimization problem
\begin{equation}\label{e4.17}
{\rm Min}\{\overline G(y,y_1,u) ;\,(y,y_1,u)\in\mathcal{H}\times L^2({\Omega};{\mathbb{R}}^d)\times\mathcal{X},\ \mbox{subject to \eqref{e4.16}}\}\end{equation}
has a unique solution $(y^*,y^*_1)\in \mathcal{H}\times L^2({\Omega};{\mathbb{R}}^d)$ satisfying \eqref{e4.16}. The process $X^*=e^Wy^*$ is  the solution to the variational solution to \eqref{e4.15}.\end{theorem}

\begin{remark}\rm Since $y^*\in BV([0,T];{\mathbb{R}}^d)$  and, as seen by \eqref{e4.16}, the singular  measure $\zeta_s=u_s\ne0$, it follows that the process $X^*$ is not pathwise continuous on $[0,T]$. However, by the Lebesgue decomposition, we have, $\mathbb{P}\mbox{-a.s.}$,
 $X^*(t)=X^*_a(t)+X^*_1(t)+X^*_2(t),\ \forall  t\in[0,T],$
where $t\to X^*_a(t)e^{-W(t)}$ is absolutely continuous, $X^*_1$ is a jump function and $X^*_2$ is a singular function, that is, $X^*_2=e^Wy_2$, where $\frac{dy_2}{dt}=0$. a.e.\end{remark}

\noindent {\bf Acknowledgement.} This work was supported by the DFG through CRC~1283. V. Barbu was also partially supported by CNCS-UEFISCDI (Romania) through the project PN-III-P4-ID-PCE-2016-0011.

redpajama_set_name: RedPajamaArXiv


On July 17, various groups joined together for a working meeting to discuss the growth of the non-native Phragmites australis, or Common Reed, in North Carolina's coastal marshes.
Phragmites grows very quickly and can diminish a marsh's productivity. Photo by Mark Hibbs.
The workshop, organized by the North Carolina Coastal Federation, was held at the National Oceanic and Atmospheric Administration (NOAA) lab in Beaufort, in partnership with the N.C. Coastal Reserve & National Estuarine Research Reserve (NCNERR) and the Albemarle-Pamlico National Estuary Partnership (APNEP). The meeting brought together researchers, policy, agency staff, landowners and others with expertise of this invasive species. The goal was to evaluate what is known about the science and management of Phragmites, and to discuss the development of a coastwide management plan to better manage the spread of the plant.
Nationally, current management programs point out the difficulty and complexity in managing this species. Once Phragmites becomes firmly rooted, aggressively spreads and grows up to 12 feet high, it can diminish a marsh's natural productivity and habitat diversity, and be nearly impossible to eradicate.
Morgan Rudd, summer coastal policy and restoration intern with the federation, gave a presentation about the current science on Phragmites. She found that 93 percent of Phragmites invasion is driven by disturbance during shoreline development. She presented findings on the ecologic impact of Phragmites within our coastal areas, including both positive and negative impacts of its growth on the native plants and animals, especially macroinvertebrates.
Ann Daisey, community conservationist with the federation and the Dare Soil and Water Conservation District, presented on current strategies for managing Phragmites. Techniques include goat grazing, mowing, cutting, burning, biological and chemical treatments. Some eradication techniques temporarily control the size of the plant so that other plants have room to grow, while others kill its rhizomes, preventing it from spreading further. Current management strategies are often combined, with the majority demonstrating successes in managing the growth of the species with early intervention, but few report total eradication of the populations. Karin Kettenring and Christine Rohal from Utah State University presented their findings on best management strategies, reporting on the most effective combinations of approaches and timelines for treatments.
Chemical herbicide controls dominate current management strategies by inhibiting the enzymes needed for the plant growth. While successful in killing both above- and below-ground growth, the effect of these pesticides on the ecosystem and human health is currently an issue of considerable research and debate.
Morgan Rudd, summer intern with the North Carolina Coastal Federation, presented on the current science on Phragmites australis. Rudd found that 93 percent of Phragmites invasion is driven by shoreline development.
Bree Tillett, coastal specialist for the federation, touched on current mapping of Phragmites. Mapping is available for individual sites, but the maps generally utilize imagery that is outdated. She also presented on various types of mapping techniques that could be utilized, including U.S. Geological Survey and U.S. Army Corps of Engineers mapping, remote sensing, citizen reporting and drones.
Rob Emens with the N.C. Division of Water Resources Aquatic Weed Control Program talked about the current aquatic weed control program and history of Phragmites control in the state. Phragmites is on the North Carolina State Noxious Weeds List, but not on the federal list, allowing it to be taken across state lines. He also reviewed ways that local governments can receive assistance with aquatic weed problems, such as Phragmites.
Tim Ellis, ecosystems analyst for APNEP, presented on potential collaborative ways to address the invasive plant. Many of the workshop participants agreed that working with APNEP to come up with a pilot plan to address it in the Albemarle-Pamlico peninsula would be an effective first step.
Concluding the day-long meeting, the participants generally agreed that more effort was needed to develop an effective management strategy for Phragmites australis. Future needs were identified, including a solid mapping effort, more North Carolina-specific scientific studies, methods to educate property owners on early identification and treatments, a better understanding of the effect of the plant community on the coastal region and the effects of the current management approaches on the ecology and on human health. Next steps include identifying partners for a working group, obtaining additional funding sources, and creating a pilot management program.
To stay updated on the efforts to understand Phragmites australis in North Carolina, please visit nccoast.org/invasive.
Click here to download a .zip file of the workshop presentations.


The SPC blog: Reverse payments: (how) do they affect SPCs?
Reverse payments: (how) do they affect SPCs?
While discussing possible subjects for discussion at The SPC Blog's second annual seminar this coming January, one of the topics that was raised was that of "reverse payments". The idea of a patent owner "buying off" a generic competitor in order to ensure a longer period of exclusivity and avoiding a legal challenge is one that has become well enshrined in US patent practice -- and although concern has been expressed by antitrust regulators, consumer groups and others, the Supreme Court has yet to agree to hear an appeal on this issue.
Have reverse payments become part of European practice and, if so, to what extent? There is certainly some support for the proposition that they are illegal in their basic form. But would they especially nuanced where the right which the proprietor seeks to protect lies in the realm of supplementary protection? I've not been able to find a literature on this subject, but would like to consider it as a seminar topic if there's enough substance to it.
Please let me know your thoughts, either by posting them below so that everyone can see them, or by emailing them here. If there are other topics that appeal to you, please let me know about those too.