LONDON—The dollar fell to its lowest in almost two weeks on Wednesday and the euro gained, with currency traders optimistic about peace talks in Ukraine, even amid warnings about the damage to Europe’s economy. Equity markets were up in the Asian session, continuing a pick-up in sentiment on Wall Street as markets became hopeful that the Ukraine conflict could end— although this move ran out of steam as European shares opened in the red. Russia promised on Tuesday to reduce its attack on Kyiv, but the United States said it had not seen “signs of real seriousness” from Russia in pursuing peace. The dollar extended its losses on Wednesday, hitting a new thirteen-day low of 97.797, as investors changed their mind on their defensive bets. At 1107 GMT, the dollar index was down 0.4 percent on the day at 97.993. The euro rallied against the dollar, with the pair up …